Can you please point to alternative data storage tools and give good reasons to use them instead of good-old relational databases? In my opinion, most applications rarely us
I would offer RDBMS :) If you do not wont to have troubles with set up/administration go for SQLite. Built in RDBMS with full SQL support. It even allows you to store any type of data in any column.
Main advantage against for example log file: If you have huge one, how are you going to search in it? With SQL engine you just create index and speed up operation dramatically.
About full text search: SQLite has modules for full text search too..
Just enjoy nice standard interface to your data :)
Object databases are not relational databases. They can be really handy if you just want to stuff some objects in a database. They also support versioning and modify classes for objects that already exist in the database. db4o is the first one that comes to mind.
If you don't need ACID, you probably don't need the overhead of an RDBMS. So, determine whether you need that first. Most of the non-RDBMS answers provided here do not provide ACID.
In some cases (financial market data and process control for example) you might need to use a real-time database rather than a RDBMS. See wiki link
You can go a long way just using files stored in the file system. RDBMSs are getting better at handling blobs, but this can be a natural way to handle image data and the like, particularly if the queries are simple (enumerating and selecting individual items.)
Other things that don't fit very well in a RDBMS are hierarchical data structures and I'm guessing geospatial data and 3D models aren't that easy to work with either.
Services like Amazon S3 provide simpler storage models (key->value) that don't support SQL. Scalability is the key there.
Excel files can be useful too, particularly if users need to be able to manipulate the data in a familiar environment and building a full application to do that isn't feasible.
BTree files are often much faster than relational databases. SQLite contains within it a BTree library which is in the public domain (as in genuinely 'public domain', not using the term loosely).
Frankly though, if I wanted a multi-user system I would need a lot of persuading not to use a decent server relational database.