I understand that a watchtower service for the Bitcoin lightning network requires fees to do their job. How much is this fee to justify a 24/7 server uptime cost? Can one use the bitcoin lightning network without watchtowers to avoid this fee?
It is still very early in the evolution of the Lightning Network and I don't think anyone, watchtowers or even routing nodes, is making that much money at the time of writing. Talaia Labs is working on Eye of Satoshi and it is listed as a c-lightning plugin. There are different models, perhaps you don't outsource the watchtower to a third party and run one yourself on a separate more reliable server. There are also different ways to compensate a third party watchtower e.g. a monthly/yearly fee or a percentage of the justice transaction if they need to broadcast it. On top of that eltoo (which needs a consensus change, a new sighash flag ANYPREVOUT) makes the operation of a watchtower simpler as it only needs to store a single justice transaction rather than a different justice transaction for each prior revoked state. However, the incentive to run a watchtower or outsource a watchtower to a third party definitely exists today especially as the funds you have in your Lightning channel increases.
Can one use the bitcoin lightning network without watchtowers to avoid this fee?
You can and I suspect most Lightning nodes are running without watchtowers today but it is taking a risk that your channel counterparty tries to close your channel with a revoked state during a period when you are offline. Some nodes will be comfortable with that risk, if they know and partially trust their counterparty or have near perfect uptime, others nodes will be less comfortable with that risk.