This is a complex question; I'll try and explain a bit without wandering too far off into the weeds.
First, we have to ask "what is true randomness"? Such discussions quickly degenerate into philosophical waters, but the gist is this: "is the universe truly random"? In other words, if you quantize time and matter, can you compute the next state of the universe from the current one? If yes, then the universe is deterministic and there is no true randomness (see what I mean about "philosophical"?)
Because "true randomness" is difficult to define, we often settle for "pseudorandomness." This is generally required when generating "random" numbers on a computer, of course.
The simplest pseudorandom number generator would be something like Dilbert's famous "9.. 9.. 9.." algorithm. But intuitively it doesn't seem very good (which of course is the joke). Statisticians have developed a whole host of tests to say whether a sequence of purportedly random outputs are "good". Start with the wikipedia page for "chi squared test" and you could spend an afternoon just reading about these tests.
A simple computer algorithm like a "linear congruential generator" produces numbers good enough for a chi-squared test (you still need to "seed" this algorithm from something, however).
The next step up in "goodness" is "cryptographically strong randomness" which means that given a sequence a1, a2, ... you cannot predict the next number in the sequence with "reasonable probability" unless you use a lot of computation. These numbers are sometimes called "computationally pseudorandom." One common way to obtain such a sequence is via a "hash chain" like this: a1 = SHA512(a2), a2=SHA512(a3), ... Since we believe (based on experience, not mathematical proof) that SHA512 is computationally hard-to-invert, we believe that a2 is "impossible" to predict given just a1.
So now the question arises, what's the best thing humans can do under the rules stipulated in your question? Humans are notoriously bad at generating randomness; there used to be a web site that would have you attempt to generate coin flips by "randomly" typing H, T, T, H, H, T, T, etc. as if you were flipping a coin (but you do it in your head). After a while, the web site would start to predict your flips better than 50% of the time (using a Hidden Markov Model). We are just bad at this.
There are ways to improve the situation using various mixing techniques that are probably doable in your head. And there are even applications I could dream up for why you might want this (political prisoner wants to encrypt a message to outside allies). But I think this post is long enough. :)