It's a bit of a loaded question, but generally it's going to depend more on environmental limitations than anything with Craft itself.
If you're on a shared host along with 200 other accounts using the same MySQL server, then you'll likely notice slowdowns sooner than later.
If you're on a decent host with plenty of memory/disk/CPU available, then we've seen sites with 500k users/entries/assets/etc. running along just fine.
And is there perhaps a way to automatically delete entries after x amount of time?
As Andrew pointed out in the comments, you could write a simple plugin that exposes a controller action that gets all entries over a certain age and deletes them. Then you'd setup a CRON job to call that controller endpoint on whatever schedule you want.