Ethernet over residential cat3 is often viable. Many homes have cat3 wiring for telephone. The spec for gigE says you need cat5 (maybe cat5e) for running 100m in dense bundles; shorter links in single runs can work on lesser cable.
Telephone wiring is pretty tolerant, so you might have daisy chain, star/home run, or a cursed mix. With star wiring, you just put rj45 ports on both ends, and a ethernet switch at the nexus point (which hopefully isn't outside, but often is). For daisy chain, you can put an upstream and a downstream port in each room where the connection chains, and either place a switch there if you've got wired devices for that room, or a short patch cable if not.
For a cursed mix, you've got to find the splice points, and break apart and have one port for each direction.
Depending on your wiring and your immediate connectivity needs, you might be able to do this a little bit at a time. There are also some in-wall access points with ethernet switches which could be worth considering, if you need a switch in each room anyway.
Eh. I had a run that was a mix of cat5e and cat3, and it ran gigE just fine. About 50 ft cat5e, 10 feet cat3; it was in crowded conduit to an out building, no sense trying to pull something new, when it cost zero to try what was already there. And it worked, so great.
Telephone wiring is pretty tolerant, so you might have daisy chain, star/home run, or a cursed mix. With star wiring, you just put rj45 ports on both ends, and a ethernet switch at the nexus point (which hopefully isn't outside, but often is). For daisy chain, you can put an upstream and a downstream port in each room where the connection chains, and either place a switch there if you've got wired devices for that room, or a short patch cable if not.
For a cursed mix, you've got to find the splice points, and break apart and have one port for each direction.
Depending on your wiring and your immediate connectivity needs, you might be able to do this a little bit at a time. There are also some in-wall access points with ethernet switches which could be worth considering, if you need a switch in each room anyway.