Contacts without Accounts: Is it possible, and if not, why?

Your understanding is correct; SFDC unashamedly assumes you're using a standard Sales-y workflow with Accounts and Contacts.

I'm not sure there's a really good alternative here. Some options might be:

• You could use a single master Account record with all of your Contacts as children. However this breaks down at volume (100k+ records).

• Person Accounts. Here be dragons. Do not tread here if you can possibly avoid it.

• A custom object. This gives you a great deal of flexibility but unfortunately means you can't use certain standard functionality that depends on the Contact record or whoId (mail merges, tasks, etc)

Is there a existing natural way to deal with contacts without accounts that would not require custom fields, reports, etc?

@Jonathon Hersh is spot on with his answer regarding how Salesforce is designed to work. That being said, there are some "creative workarounds" for various situations that are possible.

If you have certain "types" of contacts that don't need accounts associated with them, why not create a RecordType and assign all of them to the same "dummy account"? A client I work with does this for all of the individual contractors they use that aren't customer accounts or paid through their normal accounts receivables. You could easily create a variety of different RecordTypes for different types of contacts with each RecordType automatically being assigned to a single account you've set up for that type.

With the new Actions Feature, something like this should be extremely easy to implement and user friendly since the Account and other details would already have the majority of the fields filled-in for users.