Is There a Company That Allows Me to Transfer My Home Number So That It Just Goes to a Voicemail Service?

VoIP is the only way to do it, without paying an arm and a leg.

It is also possible to do it with VoIP without having to use any VoIP equipment, and no contracts, no commitments, and no gimmicks; just pay-as-you-go.
It is easy to do and setup yourself. Here is how:

Just sign up with a pay-as-you-go (no contracts) VoIP carrier like CallCentric. Setting up an account is free. Have your phone number ported to their service (I have done this). That costs $25 porting fee.

Or, to save the cost of porting, just get a new number for $1.95/month + 1.5¢/min incoming calls. Or get a phone number with Unlimited incoming calls for $5.95/month. (VoIP phone numbers are called DID's - Direct Inbound Dialing).

Then just set your account to direct all incoming calls to VoiceMail. Currently, VoiceMail is a free add-on. Access your voicemails online through your account, or have all VoiceMails forwarded as an attachment to your E-mail address (free).

If at some point down the road you get another landline or cell, then just set your CallCentric account to Forward all calls to your landline or cell number. It's that easy. Forwarding calls only cost 1.98¢/min to anywhere in USA/Canada.

P.S.
Just remember when porting a phone number (to Any service), you have to initiate the porting process through your New service provider (not the old provider). If you disconnect your current phone service before successfully porting to the new service, you will likely lose the old phone number.