Why is an old chain unsafe? I recently took my bike in for a service and the mechanic urged me to replace my chain, potentially at great cost (might need a new wheel). He said it might jam or pop off. He couldn't explain to me why my safety might be threatened but repeatedly asserted that it would be threatened.

So, why exactly is an old stretched chain a safety issue? How exactly might I be more likely to crash because of this?
A worn, stretched chain will accelerate wear on sprocket and chainring teeth. There will then be a greater chance that the chain will slip off of and override the sprocket teeth as you apply a power stroke through the crank. When this happens the pedal and your foot suddenly slips downward, which could throw you off the bike, especially if you were off the saddle and standing on the pedals. 

A new chain is not usually regarded as a 'great cost'. They are not cheap but not expensive either. There's no reason you need a new wheel, unless there are other problems of course. You probably do need a new cassette though.

Whether you need to replace the chain and cassette depends on your situation, your bike and how you ride. If you have a quality bike, do a lot of miles and want an efficient drivetrain you absolutely should replace it. If you do a few casual miles occasionally and are not experiencing any problems you can probably get away with leaving it as it is.