Why can I have an indoor pizza/wood oven, but not a BBQ or smoker? (or can I...?)
Why pizza/wood ovens, but not BBQ/smoker?

Fire is not fundamentally a problem indoors; there are certainly safe ways to do it, like fireplaces.

The things that make fire dangerous are lack of containment and lack of ventilation coupled with significant size. If it's at all uncontained, it's a fire hazard, and if there's not enough ventilation then you can get a smoke-filled home, carbon monoxide poisoning, and all manner of unpleasant/deadly things.

Wood-fired pizza ovens are much more like fireplaces than anything else. The fire is well-contained. They have chimneys, so that all of the byproducts of the fire are safely sent outside, and new fresh air is pulled in.

Standard barbecues/grills/smokers usually fail on the ventilation aspect. They don't have a way to attach a chimney so that you can reliably send all the nasty stuff outside. They may also be insufficiently well-contained. For example, a lot of charcoal grills make it relatively easy to throw sparks outside the grill, which is pretty bad indoors.

Indoor smokers do exist, though. Generally they're either large, with serious ventilation, or they're small stovetop things that just burn a tiny amount of wood chips in a tiny volume, so it's safe without ventilation but not exactly comparable to a full smoker.

The reason the big ones need serious ventilation is that you're deliberately holding smoke (and associated gases) inside, and you can't let that slowly build up in your home. On top of that, it's at a much lower temperature, probably 225-250F compared to a wood-fired oven at probably 600-900F, so a chimney won't even be as effective because you don't have as strong convection from all that hot air trying to rise.

Can I just seal off an oven and turn it into a smoker?

No. If you don't send the smoke out through the ventilation system, it'll be going into your home. (You might have experienced this yourself, if you've ever started a fire in a fireplace and accidentally left the flue closed. It can fill the room with smoke impressively quickly.)

I know you said partially close, but if the oven hasn't been designed for it, you're just asking for trouble. You don't really have a good way to make sure that smoke (and probably carbon monoxide) is held within the oven enough to make it a good smoker, but is not pushed out into your home. This is fire/home safety. You really don't want to be asking for trouble here, and depending on where you live, it may well be illegal.

So it's really unlikely that you can even make this conversion, and if you're doing it yourself you can't trust it anyway, and the absolute best case is that you manage to somehow build yourself an indoor smoker and you've probably had to make compromises that make it not work as well as a pizza oven. So you might as well just buy an indoor smoker. (But you still might not be as happy with it as with a regular outdoor smoker - see dlb's answer.)