My chainstays have been eaten by rust, can something be done?

Summary

You can ride it as is if you are okay with the risk, repair it with carbon fibre composite either yourself or get it done by a professional, or you can have a professional repair it by welding.

Risk analysis of riding it as is:

• Has been like that for a long time and didn't break: That's really one of the strongest and easiest to verify arguments: Those holes clearly didn't appear over night, and it seems implausible that the stand that you removed could have strenghtened the chainstays a lot. The holes don't look like rust holes at all, they look like they were cut intentionally. You said in a later comment that the missing metal pieces were still there when you removed the stand - rust doesn't do semi-neat cut-outs. Rust holes would be surrounded by weakened rusty material in a large area around the holes. Probably the holes have been there ever since you got the bicycle. So this frame already passed the test of time.
• Steady state riding: The chainstays are tension loaded under normal use and if one of them rips, your bottom bracket and pedals will lower to the ground and start scraping along it, the rear wheel will quite possibly lock up and the whole rear of the bicycle will probably turn sideways to a degree because one of chainstays will rip first and the other won't necessarily follow immediately. This means the front part of the bicycle which is most important for steering and braking will stay intact and you probably won't lose control entirely. You should be able to steer (to a limited degree) and more importantly brake. Because this is a steel frame, it likely won't fail explosively, but will slowly rip and bend within several seconds. Combined with the fact that hard braking unloads the failing parts, you quite possible will be able to stop before the frame bends out of shape too much.
• Emergency braking: As mentioned in the previous section, hard braking unloads the failing parts. So if you need to emergency brake for unrelated reasons, your ability to do so should not be impacted.
• Bumpy rides: The chainstays experience the most load (tension) during bumps, when you are in the saddle, so that's when they will most likely break. You can avoid that, don't go over bumps or pothole at speed, don't drop from elevated sidewalks onto the street, etc.

In conclusion, if you are athletic, this shouldn't be a problem. Do you have experience with losing traction on the rear wheel because of a blown tire or locked up wheel? Have you flipped a trailer before, and dragged it upside down behind your bicycle for some meters before managing to stop? If you were fine in those situations, you should be fine if this frame fails.

I would be very cautious doing this risk analysis for other people like family, friends or customers (just be safe in doubt), but if you can do it for yourself, that's a legitimate choice.

Repair by welding

As mentioned by other answers: Welding is a learned craft. Even if you practice on scrap metal first, you are more likely to destroy the frame or even seemingly repair it, but in actuality weaken it much more, than to successfully repair it. You would need to have it done professionally, and remove and then put back on all the parts around the welding spots.

Repair with carbon fibre composite plastic

That would be a 10 to 20 minute job for someone experienced with the process, and cost US$10 to US$20 in material. Working with carbon fibres is actually quite easy so you can do it yourself, but the material cost will multiply because you will be buying abnormally small amounts. It would be cheaper to have it done by someone who has everything in stock, and even if they apply a 2x margin on the material cost and charge fair labour costs, it will likely still be cheaper than doing it yourself.

For doing it yourself, the hardest part would be sourcing the material:

• Fibres: Unidirectional woven band (lengthwise carbon, crosswise just something to hold it together), maybe 30mm to 50mm wide. Thickness doesn't matter much, thicker ones are usually cheaper for the same amount of fibre.
• Matching epoxy resin (two component): ideally from the same store as the fibres, or something else
  • explicitly for carbon fibre composite work
  • which cures around 20 degree Celsius
  • isn't especially toxic (i.e. the material safety sheet says something about not breathing it and working in well ventilated areas, but nothing about breathing protection equipment or similar). Then it's sufficient to work close to an open window. While curing the epoxy is wrapped in air-tight plastic (see below) and shouldn't give off any vapours, dangerous or otherwise. There should be no noticeable smell while curing, and while working only the slightest bit of smell.
  • long work period of 30 minutes or an hour or so to avoid being stressed
  • Optionally there is hobbyist epoxy resin which comes in containers with dosage 'pumps' where you can add the same numbers of pumps of both parts to the mixing container, or you can use a 0.1g resolution kitchen scale to measure the epoxy - if you don't have one already, I consider that a good investment. With a scale, you will need to mix at least a 100 times or better 200 times the resolution of the scale at once otherwise it won't be exact enough. That is with a 0.1g resolution scale, mix at least 10g to 20g at once.
• Foil to wrap around each chainstay while curing. In an ideal world you would have anti-stick coated heat-shrinkable band (and would shrink it right after applying it), but cellophane foil from the kitchen will do, too - just be aware you probably won't be able to get the bottom layer off again, so it won't look good.
• Various stuff that hopefully you already have, including scissors (household scissors should work, but might be dulled by it), gloves (nitrile household gloves work), accurate scale, mixing cups (yoghurt cups usually work, unless you are unlucky and it reacts with the resin somehow), mixing spatula (gloved finger works), etc.

The material cost for what you need would be like US$10 or 20 when bought in reasonable quantities, but because you only need such tiny amount, it would probably set you back closer to US$100.

Then read the instructions carefully before buying anything or following them:

• Sand the whole area (maybe 5cm in both directions along the length of the tube, and around the whole tube, you will probably need to go onto the bottom bracket to get 5cm clearance from the holes) until there is no rust and it's slightly rough, don't use fine sand paper, use coarse one. Steel wool might work too.
• Clean it with alcohol and make sure there is no residue - pure alcohol (ethanol or propanol or whatever, in a solution with some water is ok too) doesn't leave residue after it dries, other cleaning products might. Don't touch the area with your fingers afterwards, use gloves for the rest of the process.
• Optionally - stuff some foam (most will work) into the chainstays to fill the holes and leave a surface that is as close as possible to level with the rest of the outside of the tubes. This will prevent epoxy from going into the tubes (saving weight, which is probably not a concern) and strengthen the part with respect to forces pushing onto the surface at right angles (which hopefully also shouldn't be a concern - you can not put a clamping stand back over it in any case).
• Cut some unidirectional carbon fibre band to suitable pieces - if you have 50mm wide band then maybe 5 pieces each 10 to 15 cm long for each chainstay. Make up your mind about how to place them around the chainstays, more layers at the bottom where the hole is, but I would put some onto the top too. The fibres should align with the length of the tube because that's how they are strong, when they are loaded under tension. Don't worry if they aren't exactly aligned with the length axis of the tube, if you are off just a little that will give you some torsional strength instead of tension strength, which can't hurt.
• Prepare some plastic foil or band, see following instructions to understand what you need. Cut it into suitable pieces and lay them out so you can grab them with epoxy-stained gloves. Also prepare pieces of adhesive tape to fix the plastic foil.
• Mix epoxy resin thoroughly. This is not the place to save time.
• Dip your carbon fibre band pieces into the epoxy, get rid of excess epoxy by pulling them between your (gloved!) fingers.
• Put them onto the chainstays, they should be sticky enough to stay there on their own.
• Wrap the plastic foil/band around each of the chainstay assemblies as tight as possible, to squeeze out as much air and epoxy as possible. That's right, you want the least possible amount of epoxy remaining between the fibres. It's okay that some will go into the inside of the chainstays and stay there.
• After the epoxy has cured (follow instructions from epoxy - probably 24h at room temperature or similar, make sure you don't accidentally get epoxy that needs to be heated while curing) take off the plastic wrapping (as good as possible, if you didn't get the non-stick kind). This is most likely by far the most noxious step because the solid epoxy will splinter and get into the air. Thankfully we have this nice little pandemic so you will have plenty of suitable breathing masks. Might also be a good idea to wear some glasses. Preferably do this outside. The stuff is not poisonous, but breathing in dust (or getting it in your eyes) is never advisable.
• Paint it to make it look better and also cover exposed steel around the edges of where you worked.

Now the reinforced part of the chainstays is probably the strongest part of the bicycle. But you can't put the stand back on over the carbon fibres, they are strong only under tension and pressing onto them with a clamp is the best way to break them.