Also: Is it something that can be achieved with limited funds? I ask because I am intrigued by the obstacles that are in place.
In addition to the right software, you also need to consider physical security and legal security. You'll want to know what are the laws concerning whistleblowing and privacy in the jurisdiction where you are hosting it. The software may be great, but if you live somewhere where you can be arrested just on suspicion, or they can break in and steal you server without a warrant, then software won't do anything to protect you.
One last thing to mention. If the type of activity you plan to engage in is illegal where you live, you shouldn't post anything to Stack Exchange that links to your real identity. Ross William Ulbricht (of Silk Road infamy) was caught in part because he used his real name – briefly – on StackExchange.
If you have to ask, then you shouldn't do it.
Security is difficult to get right, and requires a fair amount of experience to recognize all the ways in which it can go wrong. And a safe-haven for whistle-blowers represents not only a clear target for determined attackers, but it also comes with very real risks to the rights, property, and personal safety of the people who use the site. Unless you are a highly-experienced security professional, you'll be doing a disservice to the people you intend to protect by setting up a service that won't be able to provide the level of safety they will expect from you.
But to answer your question anyway.....
The cost of running such a service is minimal to start with, and scales with the popularity of the site -- much like any other online service.
Your main challenges will not be significantly different from any other sensitive application; you'll need to maintain very high operational security, privilege separation, secure coding and design, and proper monitoring and controls.
Your main threats will likely be social engineering, phishing, legal measures, and direct attacks on the technology itself.
As long as you know how to secure a service, securing this type of service does not represent a significant departure.
I would see two different risk categories in a project like that:
Legal risks: First thing, where would be physically stored the information. That will highly influence the legal considerations to take into account when creating your project. If you create a legal entity to manage this system, you also want to be sure you are comfortable with the laws that apply in the place where this entity exist (e.g. say your data is located in Iceland, but your company is US based, authorities can still get your data). Another important part here would be business continuity plan / disaster recovery to manage service disruption (e.g. service provider shuts down your access, denial of service) and data corruption (i.e. get a strong backup plan and solution).
Illegal risks: Physical and logical security are a very important part as the information you will store will very likely interest various people, including some that won't really care about the legal context in order to get your data (hackers, organizations, govs etc.). You want your stuff secured as hell, and typically you don't want to reinvent the wheel for your software solutions. Go and find what similar projects are using. Be very paranoid.
Basically, you want to make an in-depth risk analysis and determine what would be the exact threats for your specific project.
Then, to address the "funding" part of your question, the answer is yes, this can be achieved with limited funds. What you need to start this project (i.e. have a first version up and running) can be done with limited funds. The hosting provider will probably be the main spending, and the software system can be build on open-source and free solutions. After that, if your project grows, you will definitely need more money, but then I guess that it means you will have some support from your "customers" and users. Note that I refer to the technical part of your project, and this doesn't include any marketing plan or anything else related to the business aspects.
The key question is whether you need to host this anonymously. If you do, then it must be a Tor hidden service. You'll need a hosting provider that's comfortable with Tor. And you'll need to pay anonymously, perhaps using throughly laundered Bitcoins. Also, you must access the server only via Tor.
You might want to host locally, for better physical security. However, consider whether you would rather have the site compromised, or be discovered in possession of it.
If you don't need anonymous hosting, you can ignore most of that ;)