This is probably the #1 use case among our users, so yes it is totally do-able. Depends a bit on the router, since we don't support the full range of OS's yet, but if it is FreeBSD-based or OpenWRT you should be good to go. Otherwise, any Linux computer in your home network can act as the router to the LAN.

I believe we actually had a feature like exit nodes before Tailscale released theirs :) Our is called an "egress gateway" and it acts in pretty much the same way.