Same here, this is happening since the last update.
I can talk to NPCs, choose a sidequest and accet it, but it does not start.

You can only have one sidequest active at a time, so that may be your issue.
If it is absolutely positively borked and you can't start any quest no matter what, type /questlist in chat. It will spit out a bunch of quest names (mouse wheel to scroll through chat BTW). From there, note the quest you'd like to start and type in /activatequest (quest name)

For example, for the "Go to Bill's house for the first time" the quest name is "quest_loneguy" so to activate it the command is /activatequest quest_loneguy

Originally posted by Agent West:

You can only have one sidequest active at a time, so that may be your issue.
If it is absolutely positively borked and you can't start any quest no matter what, type /questlist in chat. It will spit out a bunch of quest names (mouse wheel to scroll through chat BTW). From there, note the quest you'd like to start and type in /activatequest (quest name)

For example, for the "Go to Bill's house for the first time" the quest name is "quest_loneguy" so to activate it the command is /activatequest quest_loneguy

Thanks, ill try this since i don't have any sidequests activated at the moment.

Originally posted by Agent West:

You can only have one sidequest active at a time, so that may be your issue.
If it is absolutely positively borked and you can't start any quest no matter what, type /questlist in chat. It will spit out a bunch of quest names (mouse wheel to scroll through chat BTW). From there, note the quest you'd like to start and type in /activatequest (quest name)

For example, for the "Go to Bill's house for the first time" the quest name is "quest_loneguy" so to activate it the command is /activatequest quest_loneguy

Thanks for your reply, but that's not it.
I'm on my second playthrough, so I'm kinda used to the system.
On my first playthrough I did most sidequests after the main quests / stotryline, so at first I thought that was the problem, but after completing it, I still couldn't start a sidequest.
And as I said before, I can talk to NPC's, ask them if I can do anything for them and click on one of their quests and get the last bit of the conversation that follows, but the quest doesn't start.

edit: I had a sidequest open when the Jan 7th update came and was able to finnish that one, but after that nothing...

Finally had time to play again today and the problem resolved itself.