Fraction resistance and negative resistance
From a math point of view, fractional resistances (and whole number resistances) are in a sense impossibly unlikely to occur, because the irrational numbers are uncountably infinite while the rational numbers (fractions) are only countably infinite. 

However our measurements of resistors invariably approximate the resistance value with a nearby rational number. For example, if you measure 7.123 ohms you can express it as the fraction \$\frac{7123}{1000}\$. But this is almost certainly not the actual exact value of the resistance due to the limited precision of your measurement instrument.

Negative resistances can be built, for example with op-amps. However these only have linear negative resistance characteristics over a limited voltage and current range. 

There are also devices like tunnel diodes that have negative differential resistance (\$\frac{dV}{dI}\$) for some voltage and current range. 

If you want to know if it is possible to buy a resistor as a discrete component with a value that isn't a whole number, the answer is yes. It's entirely common to buy such things.

In the E-96 series (1% accurate resistors) most of the values between 1 and 100 ohms are not whole numbers. For example, the first few standard values are 1.00, 1.02, 1.05, 1.07, 1.10 ohms, etc.

You can also buy values below 1 ohm, for example 0.022 or 0.0033 ohms. These are most often used in current sensing applications where you want to be able to measure a current by the voltage it produces across this resistor, but you don't want to change the voltage reaching some load by very much.