After a bit of thought I reckon a sea-going replica could be possible, but perhaps not 100% authentic due to the many reasons others have mentioned regarding passenger comfort and modern safety regulations.
Some careful design and planning could be undertaken to give potential passengers a pretty close experience though. As stated above, not many would want to spend the entire time in a 3rd class cabin with a bunch of other people they didn't know, but what if as part of a 3 or 4 day cruise you spent a day in each class. The fine dining experience of 1st class with a steward for your every need on your first day, then on day two you get to see what it was like in third class for a single night, complete with the Irish dancing session as per the movie (which is what most people will want to relate to anyway). Modern amenities such as additional toilets and showers would perhaps have to be added here and there, but if kept to the style of the rest of the fittings it wouldn't detract from the experience much, most people wouldn't know that they weren't on the original vessel.
In terms of not enough things to do for a modern audience, I think this could be marketed as a plus. Call it a 'digital detox' - no phones or modern electronics allowed. Experience life like it was in the 1900s with less distractions and more time for yourself and your family. Have a good stock of period correct newspapers and books for people to read, along with deck games, the swimming pool, squash court, board games or just sitting on deck watching the world go by. An important part of the experience would also be a good selection of period clothing for people to wear, appropriate for whatever class they are in for the day. A surprising amount of people love playing dressups.
As for modern safety features, I don't have any experience in this area, but I think again with some careful design you could integrate a proper fire detection system, Public Address, radar and lifejackets without it being obvious. I'm sure it would be possible to build a smoke detection system that could be hidden in a chandelier, with additional fire hoses and extinguishers hidden in cupboards in the corridors. Life jackets and emergency equipment could be modern just discreetly hidden away unless required. Radar could be installed in a fake crowsnest on the foremast. Not sure about lifeboats, perhaps they could be styled to look more correct to the era. Radio would be a requirement on the bridge but could be kept in the wheelhouse with the radar display, GPS and other modern equipment as necessary (even these could be made to look Edwardian to the casual observer) whilst the navigating bridge would be 100% original.
Technical requirements, I think triple expansion engines and the turbine would be possible. A lot of people would show interest in this area. As for hand firing boilers I originally thought this might be a step too far, that it wouldn't be possible to have anyone work in these conditions. But there are plenty of small steamships where people volunteer to shovel coal. Modern PPE in the form of dust-masks and heat/fire protection would be a must though.
Could still have steam driven auxiliaries perhaps with electric backup for the super important stuff. Steam driven electrical generation could also work, but you'd probably have to generate in AC. Also replace the emergency dynamos with diesel generators.
Air-conditioning would have to be fitted in somewhere. Perhaps decommission some of the third class cabins and put the plant in there.
All this said, while technically and perhaps even economically possible, I personally do not think a real life replica Titanic should be made. In my opinion it is best to have the research, pictures and digital recreations that still allow us to build an experience of the original vessel and try and understand the tragedy that everyone on her went though.
Maybe you could build a replica and name her Olympic, but you'd probably find most people would have no idea what ship you are talking about.