Considering that airplanes are very noisy, could one routinely land them with the engines turned off, gliding onto the runway? Could the engines be started fast enough for an emergency situation?
Not feasible. A start of each engine can take 30-60 seconds or longer and would require the APU (noisy). In some airplanes you lose the packs during start so you would impact pressurization. Additionally many engines have a required time at idle after start before significant thrust is applied. Lastly, with all engines offline, the airplane will be in some sort of essential power mode and many systems will be offline or unavailable.
You might then ask that we land at idle but that can present issues with spool up time before thrust is available.
One of the primary reason we land with a high flap setting is so that we have the drag to maintain approach speeds with the engines spooled up. This gives us near immediate power if we need it to go around.
I'm sure someone else can provide historical data from when idle approaches in jets were normal and specifics into why we don't do this today.
Is it possible to land them with the engines off routinely?
Yes. Depending on the glide ratio of the aircraft, you most certainly can land them. And in flight training, we do practice this routinely, although we don't actually turn the engine off for safety reasons, merely reducing the power to idle. Technically, this is not the same thing, as the propeller is still turning and producing thrust, and of course the engine is on, producing noise, but it is certainly good to know how to land without an engine.
But I suspect you really were asking about the more important part of the question - should you, and is it safe?
Well, no, not really. You can check out this list of flights that required gliding. You'll notice that several incidents that required gliding were very succesful, and they glided for quite a long way. However, most notably for your question is the first flight on the list - a DC-3 on approach lost its engines and attempted to glide in. They failed to clear a mountain, and crashed, killing everyone aboard. If you were gliding into an airport, and suddenly needed power, it would take too long to restart.
Additionally, if weather conditions caused your first approach to be off, it would be definitely better to 'go around' than attempt to save a bad landing. Just watch the first landing of this video. It would not be possible to restart and engine before touchdown in that situation. Maybe it could be done earlier in the approach, but as the landing is a task-saturated phase of flight, you do not want to attempt it in those situations (in my training, we do not attempt to restart an engine under 2,000' AGL).
So in summary, yes it could be done, but no, you absolutely wouldn't want to do it because of the potential for needing instant power.
The other answers describe well why engine power is necessary during approach.
I would just like to add that powering off the engines would not decrease sound levels as much as you would expect. Indeed, other devices such as landing gears and high lift devices produce noise (especially slats) on the same order of magnitude as the engines and those are obviously necessary during landing.
It has been pointed out that I should back my claims with references and I agree that it is a good habit to take. Unforntunately, I have to admit that I only had some faint memories of a poster I had seen somewhere. Some googling led to this study  which provides some hard data on the A340. I am not familiar with the study but seems that they have managed to identify different noise sources and their related noise levels as follows :
So the engines are only 3 dB louder than the landing gear themselves just 3 dB louder than the slats and flaps. (It would seem that, from the ground at least, slats are only marginally louder. So I was wrong about that, huh!) In another study  (that I am also unfamiliar with), it appears in their recommendations that the removal of a noise source only 3 dB louder than others will only result in a 3 dB reduction of the overall noise.
 Sijtsma, Pieter, and Robert Stoker. "Determination of absolute contributions of aircraft noise components using fly-over array measurements." AIAA paper 2958.2004 (2004): 10.
 Lockard, David P., and Geoffrey M. Lilley. "The airframe noise reduction challenge." NASA/TM 213013 (2004): 2004.