If ONT is installed in the basement yes a data wire can be ran back outside to Family room. Verizon now runs cat5e or cat6 to router from ONT. So if you do run the wire just make sure you are using CMX Outdoor rated cable with solid conductors not stranded. Basically the kind you buy with pre-terminated end is not outdoor rated and will usually have stranded conductors. And it won't last, the sheath will rot over time alot faster that outdoor rated. I would also run RG6 75 Ohm since you will need it if you ever get TV service.
They will also tie in you phone line either back at the NID if need be and will also use outdoor rated cable to do so if in the basement.
You can run fiber into the Family room although I would recommend against it because if anything happens to your fiber they will typically have to run the whole thing over because when splicing fiber you normally need a foot of cable from both sides.
My ONT is about 9 inches by 6 inches and an inch thick. About the size of a router. Power supply is even smaller. My ONT is inside the garage and is Alcatel 211ML is you want to look at it online. Hopefully that is the one they will use for your house.
From what I see from all your previous posts I think I would probably be easier to put the ONT outside and run short piece of cable to nid. Then run cat6 exterior to Family room and run power to unfinished part of basement. But if you do want ONT in the basement then you would just need to run the cable to nid and Family room. ONT also supposed to be grounded with 10 gauge wire which also will need to be ran out if ONT goes inside. I saw on previous posts that Verizon does prefer ONT installations inside as first choice.
I also have a question for you. You say you have data and phone jacks at every location. Are both ran with cat5e and homeran back to a termination point. If so you could convert the phone jack to data and use that to feed to the other data lines and install a data switch.