Let's multiply 0x84 and 0x5a using keyboard and monitor (paper and pen works the same):
Converting into binary you can do digit by digit independently and should get 10000100 and 01011010.
The minimal polynomial taken for the field with 256 elements is 0x11b for AES. You should read that as a bit in the position of $2^8$ is the same as 0x1b = 00011011 binary.
The first step is to create a multiplication table for one of the factors, let's say 0x84. The first row is the number in binary. From row to row you just append a zero, and drop the first digit (shifting all other digits one to the left). But if the first digit is 1 (corresponding to $2^8$ or $X^8$, if you remember that we do polynomial multiplication modulo the AES-polynomial), then you have to add 0x1b = 00011011. Adding is without carries, i.e., xor. You stop as soon you have 8 rows, as we work with 8-bit values.
00001000 + 00011011 = 00010011
00110000 + 00011011 = 00101011
If you've written down maybe two or three multiplication tables then the time it takes to write down such a table is limited by your typing/writing speed. Not much brain activity is needed for the arithmetic.
Now look at the binary representation of the other factor 0x5a = 01011010. Each of its digit corresponds to a row of the table: The last digit to the first row, the 2nd last to the second, and so on. Cross out every row that corresponds to a digit 0, and add up each row that corresponds to a digit 1 (sorry that 5a is symmetric, for 5b = 01011011 the first row would have a 1 in front):
1: 00001000 + 00011011 = 00010011
0: 00110000 + 00011011 = 00101011
As I don't know how to cross out here, I just copied the rows with a 1 at the beginning and added up each column writing the result below the line. Adding is again without carry: Write 0 if you see an even number of 1s above, and 1 if it's an odd number of 1s in the column.
Convert to hex 10010001 = 0x91 to get the same result as Ilmari with his logarithm tables.
If you multiply several times with the same number, of course you can recycle the multiplication tables.