It's easy to look at pictures of hard drive caddies and storage arrays but that isn't going to help. As I'm sure you know, it's not just about getting a large amount of disks and throwing them into a rack - you need to think about how they will be accessed, monitored, controlled, etc. I'm also a little confused - in your question title you talk about "many" hard drives and in the detail you talk about 4 drives - do you literally mean 4 drives, or do you mean 4 drive chassis of the sort in your picture?
The most "economical way of getting lots of disk into a rack mount" is difficult to answer because what that actually means changes depending on the problem you're trying to solve. You need to define what you're going to use them for, what sort of risks are acceptable to you and how you define "economical". And while you might have a tight budget, which is fine, you need to accept there will be some real costs here, either in time or money if not both.
What sort of problem are you trying to solve
In other words, what do you want to do with the disks, how will they connect to the thing(s) you want to use them with, etc. Different types of storage are suited to different types of job - broadly speaking you can divide "a bunch of disks in a rack" into 3 broad categories depending on what they are connected to (there are lots of other ways to group this and break it down, of course)
Direct Attached storage - DAS for short.
This is a dedicated storage array that plugs directly into an already existing server to expand the storage available on that server, usually via either SCSI, (more recently) SAS, or (typically at the lower end) SATA. This will give you a reasonably economical way of providing a lot of storage to one machine. That one machine might then act as a file server and publish shares on your network to contain files, and you can even find software to turn this hypothetical server into a NAS (see openfiler or FreeNAS for examples) or SAN (openfiler is an obvious example).
Network Attached Storage - NAS for short
A NAS is essentially a minimalistic server that is dedicated to providing shared storage on a network. Typically this will be an appliance with a highly tuned OS and file system, designed to publish fileshares on a network with reasonable performance and security, and not do a lot else (though many home/small office NASes do other tricks as well).
If you're trying to provide bulk "network" storage, perhaps centralised storage for office workers to store documents, or even for their workstations to be backed up to a central point this can be a good bet. You will probably find that a NAS might cost more than a DAS solution, but then you don't also have to provide a server and spend time configuring the server as a file server. You pays your money and you takes your choice. There are some cheap NAS devices out there (like this one) but once you start talking about rack mountable devices you're talking about "enterprise computing" and the prices and features start going up accordingly.
Storage Area Network - SAN for short
A SAN is a more specialised network file store, which is designed to allow its storage to be divided between several servers and viewed "logically" on each server as if it were a local direct attached/internal device.
SANs are connected to the servers using them by a "network" that is usually (but not always) dedicated to the SAN connections to ensure both good security, reliability and performance.
SAN infrastructure and disk typically ranges from "quite expensive" to "Is that really a price, or an international phone number" so with your worries about budget you probably see this outside of your price range - though depending on your requirements it may turn out this is what you need, in which case you may be able to set one up for "free" using the resources I suggest above.
Risk, and how you define "economical"
You mention a NAS that supports RAID as being out of your price range in your question, but you need to think about risk - only you can define what chances you are prepared to take with your data and how valuable it is, but you need to be aware that the more disks you have in a storage array, the greater the chances that one will fail and the greater the chances that another will fail while the first one has not yet been replaced and brought back online. There's a discussion about this here.
This bring us to "economical" - do you consider this to mean you want the cheapest possible solution, period (which will probably be a server with a lot of DAS boxes, configured in one giant RAID 5 array) and damn the problems and risks this might bring? Or do you consider "economical" to mean "good value for money" (which isn't always the same as 'cheapest'). I'm a lot more comfortable with that second definition myself.
Other considerations
If you want a rack full of hard drives, then you need to be aware that this will require a good power supply and will also generate a lot of heat which will need to be removed/cooled in order to keep the hard disks operating reliably, so air conditioning and careful planning of rack air-flow and power needs may be a requirement.
Unless you have a very good reason - density, etc - I would advise against going with blades.
A good 2U server from HP or Dell would provide all you need in the way of RAM. I personally prefer HP DL380's - I have several with 72+ GB of RAM. You really need to get a better grip on the fundamentals before you start worrying about what kind of hardware to purchase. You need to have reliable power, cooling, security (locked rack / server room), network access, ram and disk specifications, etc before you start looking at the kind of servers to buy.
EDIT - there is no such thing as an all in one guide to servers. I'll provide you with some preliminary stuff to get started. There is a little bit of extra information in here just in case anyone comes across this at a later point in time.
This Tech Republic article does well describing the physical requirements you should be thinking about.
Your existing bandwidth requirements should be pretty easy to determine. From the tone of your question you have a hosted solution somewhere else, either a VPS or some kind of hosting provider. You should be able to locate your existing utilization data. Expect to provide the same amount of bandwidth or higher for your in house server. The same can be said regarding the amount of disk space you require.
You definitely need to have a UPS in place for your server. Without power conditioning in place you are asking for disaster. What happens if the power flickers for one second Friday night? Your website will be out of commission until someone notices Monday morning.
Regarding your disks, you need to have RAID in your server. I suggest either RAID 5, RAID 10 or RAID 6, See here and here. Most any modern server provides this capacity. Consult the server manual for how to configure the RAID as it varies widely by manufacturer.
There are a couple more advanced points associated with running your own server that should be considered as well.
Along with running any server the burden of maintaining backups becomes yours as well. It sounds like this is something that you haven't considered. In this situation you might go with a tape drive attached directly to your server.. In any case it is something you should be thinking about.
Any internet facing server creates some risk for the network it sits on. Use a firewall to protect it from most internet traffic. To minimize risk to The use of a DMZ is highly recommended.
Best Answer
I've always done them horizontally, because that way you always have clear access to the tang with a screwdriver. If you install them vertically there is potential for good access to the tang to be obscured by equipment above or below, or the top or bottom of the rack.