Yes, objects and arrays have storage limits. They are sufficiently large to be, for most purposes, theoretical. You will be more limited by the VM than the language.
Arrays have a limit of 4.2 billion items, shown in the spec at 18.104.22.168, for example. This is caused by the length being a 32-bit counter. Assuming each element is a single integer, that allows you to store 16GB of numeric data in a single array.
The semantics on objects are more complex, but most functions to work with objects end up using arrays, so you're limited to 4.2 billion keys in most practical scenarios. Again, that's over 16GB of data, not counting the overhead to keep references.
The VM, and probably the garbage collector, will start to hang for long periods of time well before you get near the limits of the language. Some implementations will have smaller limits, especially older ones or interpreters. Since the JS spec does not specify minimum limits in most cases, those may be implementation-defined and could be much lower (this question on the maximum number of arguments discusses that).
With a good optimizing VM that tries to track the structures you use, at that size, will cause enough overhead that the VM will probably fall back to using maps for your objects (it's theoretically possible to define a struct representing that much data, but not terribly practical). Maps have a small amount of overhead and lookup times get longer as size increases, so you will see performance implications: just not at any reasonable object size.
If you run into another limit, I suspect it will be 65k elements (2^16), as discussed in this answer. Finding an implementation that supports less than 65k elements seems unlikely, as most browsers were written after 32 bit architectures became the norm.
Source: Stack Overflow