PS4 is to weak to emulate PS3. So no, it doesn't and never will. Instead they offer streaming service called PlayStation Now. But f* the streaming if You ask me. ;]
Actually, there is a lot of PSX, PS2 and PSP titles in PlayStation Store for PS4 but... now, emulation is transparent for the user. Emulators are per game (like i.e Patapons) or game packs (like i.e Mega Man collections). There is no longer "global emulators" in firmware and categories in store with games which could be run on them... Sony did this because peoples are stupid and more often buying retro games if the are branded as "remaster", even if this is technically not true.
However, bright side of this, developers which buying emulators can use complicated and powerful live patching for various of things (trophies, resources exchange on the fly, "cheats" etc.) and at least in theory, optimize it for his specific title(s). It is better for developers which no longer afraid of breaking stuff in new emulators revisions, so they publishing game once and never touch the code again without afraid that Sony makes their life's harder with some update. For us, consumers this have the same amount positives like negatives.
If PS4 will be ever truly hacked and real open SDK developed, all this issues will fade with i.e RetroArch. Except that there is no emulator in the wild which can decrypt PSS games on the fly (i.e none of emulator support *.pbp|PSAR (homebrew PSAR's are different!), none support PS3 or PSP *.pkg). On top of that PS4 ODD doesn't support CD and DVD, only BD (but I could be wrong, never test this so I base on Sony statements which I read before PS4 premiere ;p).
Even Microsoft drop the emulation in XBO which is not backward compatible. Someone could say, but how so, while user can put disc for i.e X360 in drive and play!? That's because disc is only a validation media, while game code is recompiled natively for XBO. That's another different approach to "emulation" and Sony in PS5 took this road too! PS4 games which can be launched on PS5 needs patches, which means they executables must be replaced by PS5 native code.