According to the Wikia:
Joey and Rachel continue to date, but can't make it past kissing.
Every time Joey would slide his hand across Rachel's thigh, she would
slap it away, and Joey could not manage to get Rachel's bra open. This
is most likely due to the fact that Joey has tried many times to get
Rachel's bra off, and she got used to slapping it away; Rachel herself
notes that she was unable to stop herself thinking about the fact that
Joey was the one touching her.
After trying again and injuring Joey,
the two decide that they are not compatible with each other. They
decide that they are too good as friends to pursue a romantic
relationship and leave it at that, their relationship returning to
They broke up in The One With Ross' Tan (S10E03)
The break-up is implied in the final scene. It starts out by them asking Chandler about his relationship with Monica (source of this transcript):
Rachel: Hi! Hey, listen, can we ask you a question? When you and
Monica first hooked up, was it weird going from friends to... more
Chandler: Kinda... you know, sneaking around, having to hide from you
Rachel: No, no, no... No, I mean... se-x-u-ally...
Joey: Yeah, was there a part of you that... felt like it was... really
Chandler: Actually, no. No, it felt right. You know, it felt like
uhm... I can't believe we haven't been doing this the whole time.
(Rachel and Joey are still looking at Chandler, slowly letting his
words get to them)
Chandler: I can tell from your expressions that that's the good news
you were hoping for... Well, I'm gonna go continue to... spread the
joy.(Chandler leaves the apartment. Joey sighs)
They then apply this to their relationship:
Rachel: Well, just because it happened that way for them doesn't mean
it has to happen that way for us.
Joey: Yeah, yeah... Absolutely. I mean, just because something's
difficult doesn't mean that you quit.
Rachel: Right, totally.
Joey: Yeah, so we just keep trying and trying until we... do it.
Rachel: Yeah, and if doesn't work, then we'll be just one of those
couples that never have sex.
Joey: That's a... pla-an.
And that's the point where they realize their relationship doesn't work.
(They both stare for a while, and then look at each other)
Joey: (sighs) Wow... I did not see this coming.
Rachel: I know.
Joey: I don't get it. I mean, I was so sure this was what I wanted.
Rachel: Hmmm... Me too...
They've resigned themselves to the failure.
(She puts her head on Joey's shoulder and Joey kisses her on her
Rachel: I wonder how Monica and Chandler could do it?
Joey: I guess they weren't as good friends as we are.
(Lifts her head from Joey's shoulder)
Rachel: Aah... I bet you're right.
And then they finalize their break-up, without actually expressing it:
(They look at each other for a while)