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 :
- Engines ~ 130 dB
- Landing gear ~ 127 dB
- Slats, flaps ~ 124 dB
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.
There is one thing that previous answers completely missed. It is the simultaneous controlability of descent slope and descent speed. Any glider pilot (I am one of them) knows that there is a strong relationship betweeen slope and speed. This is why almost every glider have airbrakes, they allow some independant control of both parameters. For this the approach in a glider is made at a glide slope near 10:1 although except some very old gliders most of them have a best glide ratio higher than 30:1, which would be completely unpratical for landing. The combined action on airbrakes and pitch attitude with the elevator can be used to change airspeed on a fixed slope or to change slope while keeping the same airspeed, as long as the slope is high compared to best glide.
On a powered aircraft this control is obtained by adjusting power and pitch attitude. This requires the opposite to the glider case : the approach should be on a slope less than the best glide slope, the standard is 5%. This corresponds to nearly the best glide of a liner, so no correction would be possible. Of course an unpowered approach could and should be on a higher slope, but the lack of independance between slope and speed remains.
I know of 2 cases where the problem occured.
The fist one is known as the Gimli glider, a Canadian 767 that went out of fuel due to some miscalculation on the fuel at takeoff and landed on an unused airfield using slip instead of airbrakes to adjust the glide slope, as was done in older gliders with unefficient or no airbrakes, one of the pilots was an experienced glider pilot knowing this technique.
The second one is probably the famous case of Sullenberger landing an Airbus on the Hudson river. Some people argued that he was in gliding range of two airfields when he took this decision. But the lack of possibility of precise control of the approach slope toward a runway with high buildings in the vicinity made probably the Hudson a better choice.
The start-up sequence of some engines includes a period of increased drag. This drag can ultimately crash an aircraft that is already short of power and trying to start an engine at a low altitude to get extra power. This example was a turboprop. The effect is caused by propeller positioning optimized for minimizing drag while in the off mode. Turbojet engines might not have this problem but they still take very long to start, and in fact, even to boost power in an emergency.