In truth, any ship can be made to serve as a Ratting ship, although the nature of our service often leads to certain types of ships being preferred. The most frequent ships to be found in the hands of Rats are the Asp Explorer, the Krait Phantom, the Anaconda and the Mandalay(live), due to their superior jump range and flexible modules. There are a few things that come as a bare minimum to enable your ship to refuel another though:

Jump Range: Because Rats are assigned based on how quickly they can reach a client, maximizing jump range is a good idea. We highly recommend a minimum 20Ly jump range before going on rescues or requesting a drill. A more reasonable range for 3-5 jumps to a typical case in the bubble is 40ly.
Fuel Scoop: Our clients often end up in uninhabited space, and every limpet you give to the client takes 1 ton of fuel out of your tank. Without a fuel scoop … you will soon find yourself in serious trouble.
Fuel Transfer Limpet Controller: This is what programs, launches and guides the limpets on their way to the client. Class ‘A’ controllers are preferred for their range, which can make a difference if the client is moving for some reason or if you get dropped further from the client.
Limpets: All is nice and well, yet without limpets, your shiny ship would be very, very useless. Don’t forget to resupply your stock through [ADVANCED MAINTENANCE → RESTOCK] at a station’s menu. You should carry at least 8 limpets, although consider that a reasonable minimum. Ships with a class 7 FSD, for example, can be very fuel hungry – twelve limpets might only just be enough to get such a client fuel for one jump. In a pinch limpets can be synthesized: 10 Iron + 10 Nickel for 4 limpets.
Cargo Hold: Limpets need space to be stored in, so you need a cargo hold with enough storage space (1t per limpet). It's recommended that a minimum of 8 limpets are carried. Most Rats carry 12-16 limpets. Remember that too many limpets will lower your jump range.

Some equipment is optional, but can still be very useful in certain situations:

Supercruise Overcharge FSD This is only available on live game modes. But since it is a straight upgrade over the regular FSD, there is no downside to installing it. This module will allow you to reach clients that are 100 kls away in seconds which would require multiple minutes before. Beware the increased fuel drain rate. Some ships are more optimized for SCO than others with respect to fuel usage and cockpit shake.
Extra Fuel tanks are optional and depend mainly on the class of your Ratship. Note that increased fuel capacity reduces plottable jump range. This is most useful on an Anaconda or Mandalay built for Long Range Rescues, when using the spansh neutron plotter.
Weapons are not required. You are free to take them with you, of course, and in some cases they can save the client from hostile NPCs. However, they reduce your jump range, so consider how many you fit, and look into lightweight engineering. Consider investing in stronger defensive mechanisms such as shield boosters, chaff and/or heat sinks. We usually do not run into trouble from other CMDRs through rescues, yet it happens (very, very seldom). It's more likely that you'll run into an NPC attempting to take cargo or claim a bounty. In such situations, if an NPC becomes hostile towards the client, using your weapons to draw the NPC away to attack you instead can buy the client the time for their shields to recharge or escape.
Repair/Decontamination Limpet Controllers Occasionally you will encounter clients with low hull integrity, or they might request a repair right away. Restoring at least a bit of their hull could help them reach a station safely. Decontamination Limpets can also repair a ship, in addition they can clear the caustic status, which can be helpful when saving clients in active Thargoid areas. There is also the rescue limpet controller, it essentially trades weight and limpet range for slot efficiency.
AFMU if you build your ratship with long range rescues in mind and you intend to use supercharging from neutrons your FSD will degrade each jump. Below 80% integrity you will experience random FSD malfunctions. Use the AFMU to repair the FSD if that happens. The AFMU uses a lot of power, but can be disabled while you are not using it.

On Building a Ratship

Note that you do not need a dedicated Rat ship to do your ratting in. Many rats prefer to customize their ship for ratting by using light, class D modules and foregoing weaponry for added jump range and speed, but as long as your ship has a fuel transfer limpet controller and a few limpets, any ship in your shipyard can be used for ratting. People often rat 'on the side' while out doing other things, like mining, exploring or combat. Especially on long range exploration trips, many a client has been saved by a rat that just happened to be in the neighbourhood, rather than thousands of light years away. Be aware that having a mission active or cargo in your hold (other than limpets) brings with it the possibility of NPCs spawning on you and becoming hostile. Consider only calling jumps in such situations as a last resort.

Example Builds

Below are some examples of builds across the whole budget range. There are 2 builds per example: one Unengineered and one engineered. The engineering for these examples only need access to Felicity Farseer and include the Guardian Frameshift Drive Booster.
These examples are not fully optimized and are intended to give a nice starting point to build your own ratship.

Warning!

Some of these examples can carry more or equal amount of limpets as their fuel tank is large. This means that as a rat you can give a client your whole main tank and you will not be able to jump yourself.

Builds for Legacy Elite

The following builds are good options for Legacy PC as well as XBox and Playstation. They of course work just as well in Horizons and Odyssey.

	The Usain “Rat” Bolt – Cobra MK III: Quick and nimble. Your clients will wonder what that blur was. Budget: 6 mil cr / 9 mil c Fuel/Cargo Capacity: 16t/8t Build: Unengineered / Engineered Top Speed: 290 (415) / 375 (540) Fully laden range: 25.30LY / 49.56LY
	The Diamond in The Rough Rat – Diamondback Explorer: When you want something simple and functional and to look cool while you’re ratting. Budget: 11 mil cr / 14 mil cr Fuel/Cargo Capacity: 32t/8t Build: Unengineered / Engineered Top Speed: 255 (335) / 330 (436) Fully laden range: 36.81LY / 67.51LY
	The Road Trip Rat – Asp Explorer: Old Faithful: A great all round Rat Ship, when you're not doing Robigo that is Budget: 42 mil cr / 50 mil cr Fuel/Cargo Capacity: 32t/16t Build: Unengineered / Engineered Top Speed: 260 (355) / 340 (465) Fully laden range: 32.19LY / 61.99LY
	The Krait Full of Fuel - Krait Phantom: New and improved. A great all round Rat Ship, fast and nimble Budget: 80 mil cr / 87 mil cr Fuel/Cargo Capacity: 32t/64t Build: Unengineered / Engineered Top Speed: 250 (350) / 330 (455) Fully laden range: 29.23LY / 57.74LY
	The Rat King – Anaconda: OK, now you're just showing off... At least it isn’t gold. The Anaconda is a good option for long range rescues. Budget: 255 mil cr / 262 mil cr Fuel/Cargo Capacity: 32t/64t Build: Unengineered / Engineered Top Speed: 190 (250) / 245 (330) Fully laden range: 32.76LY / 64.85LY