Homebuilt magnetic tape head
While Theran's answer is approximately correct, it's worth pointing out just what a "small" gap means. 

For replay, the width of the gap should be no more than half a cycle of the highest frequency you want to replay, (and certainly less than a complete cycle!)  For 20kHz, that means the distance the tape moves in 25 us. At 4.75cm/s (audio cassette) that distance is 4.75e-2 * 25e-6 m = 1.2 um. For a pro audio tape machine (38cm/s) 10um would be the maximum recommended gap.

For (audio) recording, it's a bit more complex; you want a wider gap to increase the magnetic field strength you communicate to the tape. Because of the AC bias, the actual gap width doesn't affect the frequency response so much, the recording takes place at the trailing edge of the gap : tape in the gap is being continually erased and re-recorded by the bias signal, so it's only the last half cycle that matters. You can get away with gaps of 3-4um (cassette) to 30um or so (15ips professional reel-reel tapes). 

This is why all good audio tape decks have separate record and replay heads (though both may be built into the same casing). Off-tape monitoring is merely a bonus.

In short, although it may be theoretically possible, building a good one with even the best equipped home workshop would be difficult. And if you decide to go ahead, at least now you know what you are aiming for.

Also, while the gap may be 0.5 to 3mm long, if you are to play anybody else's tapes, that same tight micron tolerance must apply to straightness of the gap along its full length, and to its direction : it must be exactly at right angles to the tape (there is usually an "azimuth alignment" procedure to guarantee that). Otherwise high frequency response will be poorer than expected.