It doesn't need to be contiguous. Modern computers use something called "paging". So while the physical address of the memory isn't contiguous, the virtual addresses given to the program is contiguous and can be used as if it was contiguous, thus adverting the problem with trying to keep large chunks of memory free associated with contiguous allocation methods. I've been able to allocate arrays (and other forms of "contiguous memory") up to 3GB on my system without having any problems (the last GB was taken up by my OS/other programs).
There is one case I can think of that would cause a memory error in an ArrayList before a LinkedList:
Since an ArrayList grows by "doubling" it's size (I say doubling, though I'm not exactly what's the multiple they use. I know in Visual C++ they use a 1.4x growth), you need to temporarily have space for your original ArrayList and space for an ArrayList twice the size as the original. However, you can create an ArrayList which initially has that double capacity and you'd be fine.
A LinkedList required several pointers over an ArrayList per element (depending on what kind of linked list). For example, say you had a doubly linked list. Every node would contain 3 references: one to the previous node, one to the next node, and one to the actual data. An ArrayList only requires the reference to the data. Additionally you need the overhead for each node object. This could be about 8 bytes per node (don't quote me, I haven't looked too deep into this, it could be more or less).
edit: Your computer can easily hold even the largest books in the world in memory. At 1 doubly linked list node per character in the bible, that'd take up ~40 MB of memory, and with the "worst case" arraylist which has twice the capacity as the size, that's ~20 MB of memory for the ArrayList.
did a little more reading on paging, and it turns out that when you're using the RAM, memory addresses are contiguous. Paging takes place on an auxiliary location (for example the hard disk) and doesn't necessarily hold contiguous blocks of memory. When memory needs to be swapped in, it can be re-arranged to be contiguously in the RAM. Additionally, paging is quite slow compared to operating only on RAM.
For more info, see: Paging - Wikipedia, the free encyclopedia