Two musical tones are played at the same time. One has a wavelength of 1 meter and the other a wavelength of 1.003 meter. What is the frequency of these tones that one hears?

Well, it really depends on the speed of the sound through the air between the musical-tone-player and your ear.

I'm going to assume that the speed of sound through the air is 343 m/s, because that's a number that I saw somewhere.  And now I shall go ahead and answer the question that I just invented:

Frequency = (speed) / (wavelength)

Frequency₁  = (343 m/s) / (1 meter)  =  343.00 Hz .

Frequency₂  = (343 m/s) / (1.003 meter)  = 341.97 Hz .

The frequencies are only 1.03 Hz different.  This will be the frequency of the "beat" you hear between the two tones, but it won't sound like another separate tone. Instead, it'll sound like the volume (loudness) of the two tones is rising and falling every 1.03 seconds.  I call it "wah wah".

You may also hear another tone in the air, and you'll wonder where it came from since the player is only playing two tones but you hear three.  The mysterious third tone has a frequency of 684.97 Hz, and it's the other   beat between the two original tones, at the sum of their frequencies.

The third tone is so close to the octave above both of the original two that you probably don't even hear it as a separate distinct tone.  Instead, with all four of these sounds going on, you may only hear a nice rich sound with a soft wah-wah on it, much nicer than just a couple of clean bare tones coming out of an electronic sound generator.

And then you'll understand why every key on a piano hits either two or three strings inside, and the strings are tuned to slightly slightly slightly different frequencies.  This makes the sound of each key much 'richer' than the sound of just hitting one string.