Many desktop computers support up to 32GB of RAM. Typical users only need 4-8GB of RAM (even hard core gamers only 6-8GB). If doing video rending, going higher is better. For laptops, you can find models with 16GB of RAM.
** Edit **
Sounds like you are confusing RAM and hard drive space. I'm sure you'll want a decent amount of RAM, but it really sounds like you want a lot of hard drive space. For a laptop, you'll be a bit more limited due to the physical constraints of the laptop dimensions. That doesn't mean you can't use a laptop though, just that it complicates your configuration a bit.
Desktop - Any desktop will support a couple hard drives. This would be the easiest option since you can pick up 2 TB drives and add them to it. Any computer would work, obviously there will be better options and I can offer suggestions with a little more feedback.
Laptop - Can install a single large drive and external drives (via USB cable so this is an option for a desktop configuration too).
An alternative is to get Network Attached Storage (NAS) if you primarily intend to stay within the studio for recording. A NAS is an appliance that you can put on your network and it acts as a storage device. These normally supports RAID (Redundant Array of Insignificant Disks) configurations to allow for large amounts of storage in single volumes and can be configured for various levels of redundancy (data protection) and/or performance (striping). Without knowing your budget, I can't suggest a particular model so I'll list a few options from NewEgg:
Please post another update on your preferred option. If you already have a PC and just want to increase storage, the NAS solution could work well for you. If you don't care for that, I can find some systems that you could add drives to and build an internal RAID configuration.
** Edit 2 **
Deleting the expensive NAS link since that probably isn't an option. Adding a few laptop links ranging in price (all from NewEgg). I'm not very familiar with the requirements for audio applications but I doubt having a high end video card is a requirement so I focused more on the CPU, RAM, and hard drive space. While people may suggest against the AMD CPU, the AMD CPU may perform better with audio applications that prefer more cores rather than less cores at a higher speed per core. For the RAM, 4-6 GB should be fine for you unless you do a lot of editing of the sound (at which point you should see what the audio software recommends). For storage, I limited the search to over 640 GB for an internal drive. The largest internal storage looked to be 1.5TB, but you may find that using the external drive (3TB linked below) may be a better option. You can store past sessions on the external and use the internal for new audio capture or carry the external with you when working away from home.
** Edit 3 **
For an internal RAID configuration, you'll be looking at a desktop computer then, not a laptop. There are multiple RAID configuration options, but the RAID 0 (striping) and RAID 5 levels. RAID 0 is commonly available on desktops that support hardware RAID configurations and you don't lose any of the drives to "parity" (see the description on the Wiki link). You'll get full access to the disk (minus the filesystem overhead you get with any disk volumes) but you won't have the fault tolerance you would get with RAID 5. If you regularly backup important files to another drive or removable media, this may be the better option for you. If data integrity is more important, using a RAID 5 configuration may make more sense. A RAID 5 configuration requires a minimum of 3 drives and 1 of those 3 drives will be used for "parity" to provide the fault tolerance. You can add more drives, but you'll always lose at least the 1 drive to parity. For a desktop system, you'll probably have SATA connections for 4-6 drives.
Of course I may be over complicating things by suggesting a solution that is beyond your needs. A 1TB drive is approximately $140 (can get some a little cheaper, but you'll want to look at hardware reviews before picking one). 2 and 3 TB drives are available as well. A 1TB drive is a lot of space and may be plenty for a non-professional recording laptop, especially if you plan to manage what data you keep and what data gets deleted (I'm sure you'll want to clear out bad recordings at some point).