In most languages, you can look at a word that you've never seen before and be able to pronounce it for the most part, but can this be done in Chinese?

No, is the short answer. You have to memorize them all.

There is some phonetic component, more applicable to the ancient form of the language when the script was being created, but still partly applicable today.

Chinese characters are often combinations of a meaning part and a pronunciation part. A loose analogy is this. There is a basic meaning part, say a picture of water, that occurs in the characters for river, sea, drink, and so on. Then (if we were doing this in English) you could represent the word 'sea' by a complex character water + eye. That is, 'sea' is the water word that sounds like an eye word, 'see'.

Whether it's actually possible to guess at modern Chinese characters from a knowledge of the phonetic components, I don't know.