Fencing seems to involve poking people with swords. Is it really safe?

Yes.  We take safety very seriously.  While the sport originated from honor duels to the death, we have moved very far away from that.  Our weapons are not sharp and the tips are designed to register a score on a machine, not to puncture an opponent.  All fencers must wear a fencing mask, designed to protect the head and face from injury.  Women fencers wear a plastic chest protector beneath their fencing jacket.  The jackets themselves are designed to withstand punctures.