Getting accepted to a funded PhD in the UK with low MSc grades I have a distinction (first-class honours) in my undergrad degree from one on the top universities in my home country, which sadly is not in the top ratings worldwide.
This followed by a MSc in the UK, in one of the top-20 world universities. My grades were very low, I failed two subjects.

I was advised to make up for it by getting research experience. In the year after getting my MSc, I got two research internships in the EU, and also a research scientist position at a university.

My goal is to get accepted to a PhD program at the University where I did my masters. I need overseas funding. Now my background is sufficient for that, and I have enough research experience to succeed. 

However, I was told that with low masters grades I have no chance to get the funding, ever. Even if I do another masters elsewhere and graduate with distinction, or get even more experience. The answer I got from my advisor was: "Studentships are allocated fairly schematically based on grades, references, publications. In order to get funding, a student has to tick all these boxes."

Is there any way to deal with this situation? Just the idea of not being able to get there ever, no matter what I do, even if I get another degree with outstanding grades, is too harsh to accept..
Short Answer: Yes you can, however it might take longer time to find a place. 

Longer Answer 

In the UK, as long as you find a supervisor that is fine with you being at his/her research group, the university does not interrupt the flow of the supervisor get you as a PhD student. 

Tricky Parts: Now you need to find a supervisor that find you suitable, please consider the following: 

What are your strongest abilities? If you failed in a subject about software architecture for example, finding a supervisor in this area is waste of time; as you can not demonstrate that you will be a good candidate for that. See in what subjects you got a good mark, or even excellent mark; go and talk to the corresponding supervisor. See if he/she knows someone so you can apply for a place. 

Supervisor First Then Funding: Based on your question, I'm assuming you are a European national. In this case, there is more likely for you to get a some European-based funding compare to Asian students for example. So, first focus on finding a supervisor, and then talk to him/her about funding; I'm sure seasoned professor, knows how to help you to apply for a funding. 

Possibilities In Smaller Towns: You could also try to find a good university in smaller towns, where the price of living is much less than big cities. Also, because the demand is lower compare to big cities, they might have a funding for you.