Why doesn't water boil in the oven? I put a pot of water in the oven at $\mathrm{500^\circ F}$ ($\mathrm{260^\circ C}$ , $\mathrm{533 K}$). Over time most of the water evaporated away but it never boiled. Why doesn't it boil?
 A: Water boils both in the oven and on the stovetop. But one is called simmer and the other is called rolling boil. What you are asking about is the visual effect that is called rolling boil, and your question is basically why does it happen on the stovetop and not in the oven.
The answer is that the oven heats up the metal pot to some lower level, not even close to the air's $\mathrm{500^\circ F}$ in your case through direct contact with the air, while the stovetop is able (through direct contact to fire) to heat up the metal pot to around $\mathrm{900^\circ F}$, which leads to faster boiling and the effect of the visible rolling boil.
https://cooking.stackexchange.com/questions/21432/what-temperatures-do-low-medium-high-on-the-stove-correspond-to
A: The water didn't evaporate. It boiled. If you could look closely at the water in the pot in the oven you would see small bubbles rising within the liquid, which would indicate boiling.  But you wouldn't necessarily observe what is sometimes referred to as a "rolling boil, i.e., large bubbles rising in the water indicating a high rate of boiling.
You get a faster boiling rate when the rate of heat transfer to the water is higher, as when you boil water on a range top set on high heat. The heat transfer rate in the oven when set on bake is much slower because it is heat transfer primarily by convection (contact with naturally moving air) as opposed to conduction (contact with a solid high temperature surface), which is a generally higher rate.
Evaporation is a different phenomenon that occurs at temperatures less than the boiling point and occurs only at the surface of the liquid.
A: Short answer is that it boils, but it boils differently because it's either evaporation from a liquid surface in low temperature or "bulk evaporation" aka. boiling, due to temperature gradient. Now because oven heats more or less uniformly all sides of pot is heated the same, thus eliminating clear temperature gradient. Without  temperature gradient "strong visual" bulk evaporation is impossible. Besides in oven air is heated to high degree, thus producing higher pressure to escaping water vapor molecules, so water becomes a little-bit super-heated, which may raise water's boiling point a bit. This is second reason why you don't see standard boiling effects as with boiling kettle.
A: The "roiling boil" is a mechanism for moving heat from the bottom of the pot to the top. You see it on the stovetop because most of the heat generally enters the liquid from a superheated surface below the pot.  But in a convection oven, whether the heat enters from above, from below, or from both equally depends on how much material you are cooking and the thermal conductivity of its container.
I had an argument about this fifteen years ago which I settled with a great kitchen experiment.  I put equal amounts of water in a black cast-iron skillet and a glass baking dish with similar horizontal areas, and put them in the same oven.  (Glass is a pretty good thermal insulator; the relative thermal conductivities and heat capacities of aluminum, stainless steel, and cast iron surprise me whenever I look them up.)  After some time, the water in the iron skillet was boiling like gangbusters, but the water in the glass was totally still.  A slight tilt of the glass dish, so that the water touched a dry surface, was met with a vigorous sizzle: the water was keeping the glass temperature below the boiling point where there was contact, but couldn't do the same for the iron.
When I pulled the two pans out of the oven, the glass pan was missing about half as much water as the iron skillet. I interpreted this to mean that boiling had taken place from the top surface only of the glass pan, but from both the top and bottom surfaces of the iron skillet.
Note that it is totally possible to get a bubbling boil from an insulating glass dish in a hot oven; the bubbles are how you know when the lasagna is ready.
(A commenter reminds me that I used the "broiler" element at the top of the oven rather than the "baking" element at the bottom of the oven, to increase the degree to which the heat came "from above." That's probably why I chose black cast iron, was to capture more of the radiant heat.)
A: It might also be prudent to consider the environment of the oven itself. The "atmosphere" in the oven is already at a temperature > 100°C and this means that water in the "air" is in the gaseous state. As water evaporates at the surface at temperatures nominally > 40°C this would mean that the water vapour is "immediately" absorped into the gaseous state (immediately in the sense that the energy transfer occurs far fast because of the molecular energy) and the remaining water is nominally cooling the surface of the vessel via thermal convection. I believe this is thermo-dynamics in action :)
A: If you include 'microwave' in the OP's ovens, the water in glass containers boils throughout its volume, not just the surface, indicating heat transfer barriers in conventional ovens.
A: In the oven, the air is saturated with water vapor, and the number of evaporated water molecules becomes equal to those settled in it. Evaporation stops.
