I'd recommend a "Virtual Private Server" (VPS). This gives you (eg) root access to a linux/apache based server that runs under a Xen share on a larger machine. You ssh in and do whatever you want. They run around $20-25 a month for a small slice, meaning you get like 8gb of storage and 128mb of RAM, sufficient to run enough apache threads to deal with mild traffic. The bandwidth will deliver to a few high speed broadband connections simultaneously. The storage is a complete normal linux filesystem that must include the OS (the experience is identical to logging in to a genuine dedicated server); for a mod perl server this is <2gb. The provider will install a minimal base distro OS (ubuntu, debian, etc -- there may be a choice) with ssh server running, then you log in and use apt or yum (or whatever you want) to install apache, mod perl, ftp, etc, etc. If you pay for an "unmanaged" VPS do not expect them to help you with any configuration, so you should be comfortable and competent with linux/apache. Again, this is identical to working remotely with root access on a dedicated server, which short of messing with the hardware, you can do everything you could do if the box were sitting in front of you. Since this is a 24/7 server, you get a dedicated permanent IP address.

I will not recommend anyone in particular but I will recommend against "VPS village/Grok.net" as I have had a very unpleasant experience with them in the past involving hardware failure and extreme negligence.