WoW Cataclysm - Is there any place to store inventory out of my bags and access it later?

Yes, there is! In any major town, you just need to visit the Bank. It should fit your needs.

According to WoWWiki, there are banks in the following cities:

Horde

1. Thunder Bluff
2. Orgrimmar
3. The Undercity
4. Silvermoon City

Alliance

1. Ironforge
2. Darnassus
3. Stormwind
4. The Exodar
5. Theramore

And for some neutral cities:

1. Ratchet
2. Gadgetzan
3. Booty Bay
4. Everlook
5. Area 52
6. Shattrath City (Aldor/Scryer)
7. Dalaran (northern/southern and sewer)

Also according to the wiki, there are portable banks as well, but I have never had any personal experience with them. You should also find the guild banks there, which are (depending on your guild setup) accessible by the whole guild.

There is also a secondary storage facility available in Stormwind and Orgrimmar called Void Storage. It requires a gold amount (100g currently) to unlock. It has certain limitations:

• Can only deposit soulbound gear
• Can only deposit non-unique gear (unique-equipped is fine, however)
• Each piece of gear you deposit costs a sum of gold (25g currently)
• Items are free to withdraw
• You can only deposit up to 12 items in one 'transaction'
• You can only withdraw up to 12 items in one 'transaction'
• Void Storage is limited to 100 slots per character.

The upside of all these caveats is:

• You may retain transmogrifications on your current gear to gear in your void storage.

Think of it as an extended bank for gear you might want to hold onto one day for looks, but won't use ever in the immediate future.

What you are looking for is the bank that supplies some initial space and the option to buy bag slots for additional space.

They are located the following places:

Horde bank branches

• Thunder Bluff
• Orgrimmar
• The Undercity
• Silvermoon City

Alliance bank branches

• Ironforge
• Darnassus
• Stormwind
• The Exodar
• Theramore

Neutral bank branches

Some neutral cities and towns have them as well:

• Ratchet
• Gadgetzan
• Booty Bay
• Everlook
• Area 52
• Shattrath City (Aldor/Scryer)
• Dalaran (northern/southern and sewer)

Addition to mobile banking:

I know of 3 ways to access your bank remotely:

1. If you are a goblin you have a racial that allows you bank access for 1 minute every 30 minutes. It's called Pack Hobgoblin (30 min cooldown) Calls in your personal servant, allowing you bank access for 1 min..
2. You are a skilled engineer and have crafted your Jeeves (link)
3. You have done Argent Tournament (link) quests (they can be started at level 78 - still some time away for you ;)). Then you can eventually buy a pony that also gives you bank access (link)