Correct usage of the word 'storage' or 'storing'?

Neither is correct, though the second sentence is half-right.

"Storage space" and "storage solution" are the usual terms used here.

Google claims 87 million results for "storage space", less than one million for "storing space"; and 12 million results for "storage solution", less than 100k for "storing solution".

You can see a graph of the popularity of these terms over time here: https://books.google.com/ngrams/graph?content=storage+solution%2Cstoring+solution%2Cstorage+space%2Cstoring+space&year_start=1800&year_end=2008&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cstorage%20solution%3B%2Cc0%3B.t1%3B%2Cstoring%20solution%3B%2Cc0%3B.t1%3B%2Cstorage%20space%3B%2Cc0%3B.t1%3B%2Cstoring%20space%3B%2Cc0

You can see from the graph that "storing solution" and "storing space" have never been commonly used.

Generally, "Storage" is the noun, the name for a container that can store things, while "storing" is the verb, what you do in a storage container.

From https://www.merriam-webster.com/dictionary/storage

storage, noun: space or a place for storing

From https://www.merriam-webster.com/dictionary/storing

store/stored/storing, verb: to place or leave in a location (such as a warehouse, library, or computer memory) for preservation or later use or disposal

There are many similar words, such as "carriage", which is the name (noun) of something that does the action (verb) of "carrying" in.

There's also signage, assemblage, shrinkage, drainage, and many others. Not all -age nouns work this way, becoming a verb when you change the ending to -ing, but many do.

Sadly, there are many other terms which don't work this way, and instead use the verb as part of a compound noun: it's "fishing pole", for example, and "chopping block", "paring knife", "dancing partner", "sewing machine"... English grammar is inconsistent like that, and I'm sorry that the only way I know of to learn the difference is to memorise by rote: I don't think there is an easy rule of thumb here.

So the "correct" version of the sentence would be, since these are both nouns:

Those who want more and more storage space for their content can understand the importance of portable storage solutions.