DubDub:
10/7 is a rational number, from the definition of a rational number as one that can be expressed as a fraction of two whole numbers. It may be infinitely long, but it's still rational.
An irrational number is one that doesn't repeat, no matter how many numbers you add to the right of the decimal point. Pi and e are the only two I can think of off hand.
You said it's possible to measure an inch between your thumb and pointer finger, and that it's not possible to do the same with a distance of 10/7ths of an inch. In theory it's possible to do either one. Think about it for a second: if you can measure an inch, you can measure a seventh of an inch. and if you can measure a seventh of an inch you can add ten of them together to get 10/7ths of an inch.
In actuality it is not possible to measure anything exactly, since a measurement of one inch is 1.0000000000000000000000 and with more zeroes all the way to infinity. You are always going to have a margin of error.
Here is how we've defined a meter over the last 200 plus years:
1793 -- 1/10,000,000 of the distance from the pole to the equator.
1795 -- Provisional meter bar constructed in brass.
1799 -- Definitive prototype meter bars constructed in platinum.
1889 -- International prototype meter bar in platinum-iridium, cross-section X.
1906 -- 1,000.000/0.64384696 wavelengths in air of the red line of the cadmium spectrum.
1960 -- 1,650,763.73 wavelengths in vacuum of the radiation corresponding to the transition between levels 2p10 and 5d5 of the krypton 86 atom.
1983 --Length traveled by light in vacuum during 1/299,792,458 of a second.
Each succeeding definition is presumably more precise than its predecessor. But each still has some margin of errorm particularly when converted to human-level neasurements. Let's look a bit further:
The second (abbreviation, s or sec) is the Standard International (SI) unit of time. One second is the time that elapses during 9,192,631,770 (9.192631770 x 10 to the ninth) cycles of the radiation produced by the transition between two levels of the cesium 133 atom.
So when we're measuring a second, we count off these little ticks and that, by definition, is a second. But how accurate can we be in converting from a time unit to a distance unit? Pretty accurate, and you could, by sheer luck, hold your fingers exactly an inch apart, but more likely something other than exactly an inch.
TEd
Edit:
Faldage posted whilst I researched my answer.
