Doesn't seem like a showstopper to me; just fall back to current behavior if str() fails or takes too long, and truncate if it's too verbose.
As for length, maybe 100 chars? I think that's around what repr() often uses.
I think Python exceptions are verbose enough, not necessary to make it more verbose, more complicated.
Do they matter in reality? No.
Make it an optional feature to be toggled with an interpreter flag. People like me can turn it on, you can keep it turned off.
I think it's pretty important that you shouldn't add features with unpredictable behaviour into the core of the language, particularly into error handling code.
Why would your code contain repr()-functions that trigger "unpredictable behavior" to begin with anyway?