For measuring offset the best bet is a circuit like this:
Does it require matched resistors?
Yup, ideally.
General rule for all
differential configurations of op amps is that resistors that share the same rĂ´le want to be matched. So in this case you want the 10R pair and the 10k pairs matched. If they're not matched it lowers the common mode rejection ratio. If you've got an theoretically perfect op amp with infinite common mode rejection ratio a mismatch of 0.1% will reduce the CMRR to 60dB, a 1% mismatch to 40dB.
What matters most here is the ratio of input to feedback resistor. If you use 1% resistors you'll get a 1% uncertainty in gain, so a 1% basic uncertainty in your measurement. For this particular application the contribution from common mode errors is less important than the contribution from gain errors - you know that you have no common mode signal on the input (it's grounded), so any common mode errors will be from bias current effects on the input resistors which will be negligible in the face of the likely range of input offset voltages. For a [huge] 10uA bias current you'll have a common mode input voltage of 100 uV reduced to 1 uV by the 40dB CMMR.
If you're dealing with ultra-precision op amps with single digit microvolt offset voltages then of course you're going to need to move to much better resistor matching but if you're paying for those kind of op amps you can afford to pay Vishay for some precision matched resistor dividers.
I'd say if you're dealing with >100uV input offset voltages and are happy with results in the region of 1% accuracy then 1% resistors will do you fine. If you're after 0.1% accuracy or dealing in smaller offset voltages then you probably want to go down the route of hand matching some sets of resistors with a good meter (Or you can pay Vishay or someone similar lots of money for top notch matched resistors with low tempcos).