IMHO I would not put my time or one cent of money into trying to get something not made for this purpose to maybe barely work. Yes only 2 pairs are needed for fast Ethernet (100Mb/s) but the standard defines a precise spec for how many twists per meter there is for each pair.
Each pair is twisted at a different rate to balance that pair to itself to minimize crosstalk between pairs.
Depending on how the phone wire is constructed and it's length there will be a distance at which so much crosstalk is introduced that you're getting an Ethernet connection, ironically, suffering from some very wifi specific problems like serious signal degradation, lag, and dropped packets.
Commercial & residential new construction & rehab work in the US has used Cat5 or minimally Cat3 for phone wiring for the last decade or so (I wired my home with Cat5e for phones which easily supports analog phone or digital VOIP in same jack (just different termination ends). So unless the wire is labeled on the outside of the jacket I would assume its old 4 conductor phone wire which isn't generally twisted (at least not for any purpose other than ease of fabrication, that I know of) and definitely not in pairs.
A few misc things in response to some other replies:
- Generally you don't want to put Cat5e in a conduit that bunches it too close to another cable or presses it tightly against a metal conduit wall. If anything, open air raceways or non metallic conduit that has 50%+ open space is preferred, and open air raceways are best.
- they make Cat5e/6 to specifically go inside HVAC ducts (google riser vs plenum cable, different states and buildings have different codes). Even POE is quite safe, voltage & amperage wise, it's quite low in the grand scheme. Those ratings have more to do with the flammability & toxic smoke inherent to different cable sheathings.
-cat5e/6 comes in both stranded & solid if you have a bending radius preference, I've always used solid for ease of punch down
Even if you can kind of make it work with that old wire, you're essentially planting minefields of unpredictability and endless headaches and money waste.
If I were you, IMHO, I'd: buy a better repeater; or get a set of power line adapters; or pay a professional to run the line you're looking for. There's no reason to Apollo 13 a situation if you aren't actually flying in outer space with an extraordinarily small set of tools and spares.
I'm glad to share pics of my own cabling if you'd find it inspirational. And I'm by no means talented.