suppose I have a table that stores 100 million records of strings of varying sizes up to 20 characters in a column field. I need to index this column, I only have a 2GB-Ram machine, is this sufficient to perform such task? Is mysql recommended db engine for storage?
If you have 2 GB of main memory, then yes, you should be able to build the index without any problems; virtual memory is a wonderful thing, and the DBMS may well arrange to spill data to disk as it goes.
If you only have 2 GB of disk space, you don't have enough space for the data and the index.
To no-one's surprise, it is 2 GB of main memory, not 2 GB of disk (that comment was mainly in jest - but these days, if someone says 256 GB, it is not clear whether they're referring to disk space or main memory; it could be either).
Yes, if the DBMS cannot create the index within that constraint, it is not worthy of being termed a DBMS.
MySQL probably can do the job. It isn't what I'd recommend, but I'm very biassed in this area as a result of being one of the developers of an alternative (commercial) DBMS. We don't have enough information about your budget etc to be able to advise reliably.
Databases are generally designed in a way that allows them to work with more data then you have available RAM. Giving it more working memory will speed things up, but it should be able to build the index and perform searches on it just fine.