I want to store a docker image with the input data, code, software, intermediate results and results. This will be a massive file. 

Is there something like the sequence archive for people who want to upload a docker image of their whole study? If not, what is the best/cheapest way to store such a file?

Any related thoughts not directly answering the question are also welcome. 
Storage of very large Docker images is potentially a bit of an issue. As Devon suggested you can check into Quay.io, but I don't know if they have any file size restrictions. In practice downloading a large single file can be a bit of a problem. You might consider separating the data storage from the docker image itself that replicates the code aspects and software. Then you could put the Docker image anywhere (Quay.io or just Docker Hub) and store all of the data in a public Amazon S3 bucket (or another cloud storage system with a nice API). The Docker Image could have code for pulling the data from AWS