Sometimes a picture tells a thousand words...
It all depends what question you ask Wolfram Alpha:
Floating point arithmetic leads to rounding errors. Non SI units are rarely defined precisely (an exception is the inch which is exactly 25.4 mm - and thus other derived units of length).
But getting back to the "what is the value" - we should go with actual units that have defined quantity (rather than the ones that are "precise to a certain number of digits"):
The pound (according to the National Bureau of Standards is defined as:
1 pound (avoirdupois)= 0.453 592 37 kilogram
defined - exact
The inch is defined as
The bar is defined as:
1 $N / m^2$
The relationship between Newton and kilogram:
g = 9.80665 $m / s^2$
defined - "exact"
Now, all you need is a calculator with infinite precision, and you can compute your number to your heart's content...
I get for 100000 psi:
100000 * 0.45359237*9.80665 / (10 * 2.54 * 2.54) =
and for 1 bar:
2.54 * 2.54 * 10 / (0.45359237*9.80665) =
comparing to your value:
As you can see, the value you got was "accurate" to all the figures you had. But I have more...
There is an often quoted paper about "What every computer scientist needs to know about floating point arithmetic". Worth reading: http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html
I say this often: you should be very, very wary of any situation in which you need to know a quantity to better than 1%, 0.1%, 0.01%... Quite likely you are tackling the problem in the wrong way and there's an elegant method waiting to be found. Except as an intellectual exercise, finding out the Nth digit of a physical quantity that isn't even an SI unit is silly. I wrote this answer just to point out that if you go back to sources, you can get the answer.
Now that's a bit silly, but since each of these values was defined, this is in principle accurate to all these (insignificant) digits. I use the word "accurate" very, very loosely.