We say Monero is scaleable because there nothing in the protocol level preventing Monero from accepting a very large number of transactions. There is no coded limit on the block size.
The processing capacity for nodes is based on a few things:
Frequency of transactions
Size of the blockchain
Regarding the first, transactions can not happen faster than nodes can accept and evaluate them. This is limited by the CPU speed, hard drive speed, and internet capacity of the node. It is very unlikely that the rate of transactions could exceed growth rates of 5MB/s, which even a 5400rpm HDD can handle. The most likely limited factor would be network bandwidth, which could make a growth rate of 1MB/s too much for people on slower connections (less than 8 mbps). Keep in mind that Monero (and Bitcoin) have growth rates well below these numbers.
Regarding the second point, this is where the argument is that the total size of the blockchain will become a major barrier to running a node. This is a much more legitimate criticism, and Monero has a larger transaction size than Bitcoin and most/all other coins. There are multiple ways that this concern can be put to ease. First, the cost of hard drives continues to fall. Right now, one can buy a 1TB hard drive that should last for the foreseeable future for less than $50 brand new. Secondly, Monero can still look into options such as pruning to reduce the size of the blockchain. There is currently little research in Monero blockchain pruning because the size of the blockchain is not much of a concern yet. In general, the size of the Monero blockchain is not a large concern for the next few years, and even if Monero becomes much more popular, it will still stay at a reasonable size for a while.
One argument is that the larger the block size, the harder it is to run a node and the more centralized the control will be. This is a valid point, but I do not think the size of the blockchain is intrusive enough to cause any real impact here.