It's not a DIY repair and is very common to many SLR's. The only suprise with your twenty year old camera is that it didn't happen sooner. The substance you can see is a symptom and not the cause which is deeper into the camera.
The cause is actually sticky shutter release magnets; over time internal lubricants from when the camera was manufactured have migrated onto the magnets and then partially dried out.
A camera repair technician usually addresses this as part of a standard CLA service (Clean, Lubricate, Adjust) which all SLR's should periodically undergo.
Although the EOS 600 was one of Canon's cheaper consumer range models (designed and built on their behalf by Kodak) it's far better than models like the EOS 300, but don't be surprised if some camera repair technicians refuse to do the work: there's a lot of delicate and twenty-year fragile plastic and electronic parts in there, most are irreplaceable if broken during repair and so many repairers will either refuse or insist that you sign a disclaimer. As a consumer SLR they EOS 600 was really designed to be a non-repairable model which would be discarded when broken or replaced entirely whilst under warranty. Many repairers will attach special conditions to the warranty on their work. Your best bet is to find a Canon specialist, if you're in the UK I can highly recommend Colchester Camera Repair Service.
A CLA will cost far more than the camera is worth, but if you decide to buy a used replacement it's also unlikely to have been serviced. But given that you can pick them up for free or next to nothing that may still be a better option for you if you fancy a gamble. Conversely, a full CLA will ensure that your camera has the best chance of working correctly for a few more years and will usually include replacement of any decaying gooey foam light seals (usually a chargeable extra, but most do it for free when I point out that I could take the job elsewhere).
In calculating whether to invest in a near-worthless camera, consider how much more you'll need to spend to find a better, freshly serviced model which will do the same job. Often the investment is worthwhile, but if you can get a better model at low cost/free (FreeCycle/Freegle) to have serviced in it's place it may be the better choice. EOS triple gigit models are consumer models, EOS double digit models are enthusiast/pro models and EOS single digit models are the top-spec models built for full professional use. You won't get a single digit model for peanuts but the double digit models don't go for much.
I hope that I've given you sufficient information on what to do next and that you're able to get shooting again soon. All I ask in return is that you take a moment to rate my answer.