Is there a solution to paying high fees when opening and closing lightning channels once we hit a fee only market? Paying several hundred dollars to open and close a lightning channel seems uneconomical. This is a common argument against lightning by big blockers, I’m yet to read a valid argument against this.
'Under the hood' of a lightning network channel open/close, a user will be sending a bitcoin transaction. In order to send a bitcoin transaction, a fee is paid to miners. 

One advantage of having a lightning channel is that you can amortize the cost of the miners fee across a potentially enormous number of payments. All else equal, having the ability to transact on the lightning network thus lowers a users expected fees/payment, small blocks or not. 

Channel factories are a technology that could allow for immense cost saving for lightning users. So this is potentially a 'solution', in the context of your question. 


  Paying several hundred dollars to open and close a lightning channel seems uneconomical. This is a common argument against lightning by big blockers


'Uneconimcal' doesn't seem like the right word to use here. If in the future a bitcoin transaction costs hundreds of dollars in fees, that would mean that the ability to send a bitcoin transaction is in high demand. By many measures, being in high demand is a mark of success, though 'big-blockers' will argue that by increasing the block size the network can increase the 'supply', thus lowering transaction fees. 

To illustrate this point, consider the absurdity of this sentence:

"That restaurant has become very popular! Nobody goes there anymore"

So you may ask "Why don't we just increase the size of the restaurant then?"

Well, the issue is that by increasing the size of the restaurant, the restaurant will lose the properties that made it popular in the first place!

So lets return to Bitcoin, to explain why increasing the block size ('making the restaurant larger') is a naïve solution that damages the network's desirable properties, while only providing a rather meagre linear scaling relief:

Implementing larger blocks means that running a full node on the network will be more resource intensive (bandwidth, computational cycles, storage, etc), and so naturally we should expect less nodes will exist due to the increased costs. This is damaging to the core properties of bitcoin (eg censorship resistance), which can only exist when the network is sufficiently decentralized. 'How decentralized is good enough?' is a question that is difficult (if not impossible) to answer, but this much is certain: increasing costs is a centralizing force, and the most conservative approach is to be 'better safe than sorry'. A blockchain network is a very expensive and inefficient way to implement a database, but the core properties mentioned above make these expenses 'worth it', at least according to the market. 



In any case, there has never been a time where the average Bitcoin transaction fee is 'hundreds of dollars'... in fact transaction fees have been quite low for the majority of the history of the network. This isn't to say that fees can't or won't increase in the future, but making changes that sacrifice the most important properties of Bitcoin seems like an extremely misguided approach to scaling the network.