Plane
Could a very fast airplane build up enough speed to escape the gravity well? Is it even possible to reach such speeds without using rockets / jet engines or explosions?
The fastest airplanes use jet (or for some experimental planes, rocket) engines. So you are specifically excepting the fastest airplanes.
A propeller-based plane is not going to be able to reach space because it works by pulling the plane forward against the air. No air and the propeller doesn't do anything. That's also a problem with a jet engine, but you excepted them anyway.
A plane also relies on Bernoulli forces to hold it up. Again, these work by manipulating the atmosphere. No atmosphere and the plane would have nothing to push against gravity.
I think that we can safely say that a plane wouldn't work.
Railgun
There is a technology called Hyperloop that people are proposing to use to travel between Earth-bound locations. It's basically a tunnel through which a vehicle is accelerated by magnetism. The escape velocity at Earth's surface is basically 11 km/s. So with an acceleration of 10 m/s, it would take about 1100 seconds to accelerate to that speed. That's a distance of 600 km.
$$d = \frac{1}{2}at^2$$
$$d = \frac{1}{2}(10 \frac{\text{m}}{\text{s}^2})(1100 \text{s})^2 \approx 600 \text{km}$$
A little application of the Pythagorean theory and other geometry and we find that this would be about 8 km deep.
$$6371 - \sqrt{6371^2 - (\frac{600}{2})^2} \approx 8$$
Making a tunnel running as deep as 8 km below the surface that is stable (does not collapse) is beyond anything that we've ever done. By contrast, the Channel Tunnel is only a quarter of a kilometer under sea level and less than 40 km long. The world's longest tunnel is the Delaware Aqueduct at 137 km. The world's deepest tunnel is the Gotthard Base Tunnel at 2.3 km (but it's in mountains).
We're not that far from it though. Perhaps in another century or two we could do it. Perhaps even earlier if it were important enough.
The maglev portion is also untested at those speeds. It seems feasible to be ready when the tunneling is.
This is basically a railgun. Putting it underground allows it to be straight so that you don't need the extra acceleration of a curve.
Alternatives in space
Once in space, they could create a space elevator. That requires creating a tether strong enough. That's another thing that we can't do now but might be able to do in a century or two. But you really need to be in space to build the space elevator. You drop the tether down to the surface. It's not something that you build from the ground up. So they need launch capability first.
Moving around in space can be done with solar sails, but this wouldn't get you into space.
Electrically powered spaceship propulsion
If none of that appeals to you, there are electrically powered forms of spacecraft propulsion that might fit your requirements.