The recent negative price of WTI crude oil futures contract indicates the receipt/storage of a barrel of crude oil can cost more than $37， which is unbelievable. Does anybody know the real cost at present, such as the rent fee of oil storage tanks?
All storage at Cushing has been contracted for. The price drop isn't just reflective of the price of storage, but primarily of the scarcity of storage. Read the Storage & Transportation section of this answer for a longer explanation.
There is no sticker price for storage at Cushing. There might have been in a different environment when it only served a logistics function, but not anymore with real money to be made. Right now, one would have to directly negotiate with an existing holder.
As for what price these holders would have paid, you should look to the shape of the crude oil forward curve for a clue. The market price for storage would primarily be driven by the level of contango (negative diff between nearer month and farther month contracts), after all, that is the value one can capture with storage.
For some actual prices, I know Cushing storage was being offered at 20-30c/bbl/month early in the year (Jan) and progressively increased to about 50c in late Mar/early Apr. I doubt there are any offers right now, but i'd guess they're priced at $1/bbl/month at a minimum considering the 1 year spread (Jun20-May21) is $12-13. Likely, they'd be priced much higher to reflect extreme market volatility and a potential for further negative price collapses as storage continues being filled.
I suspect the most economical way to store oil is to store it in a cave / tunnel. Preferably an old cave / tunnnel, because the construction costs have been already paid (or there is no construction cost if it's a natural cave). But, there is a limited number of old caves and tunnels so at some point of time new tunnels need to be constructed.
Look at https://www.researchgate.net/publication/314096492_Assessing_and_benchmarking_the_construction_cost_of_tunnels -- figure 3 has an estimated construction cost of new tunnels. The estimated minimum cost is 1500 EUR per meter per 40 m2, or 37.5 EUR / cubic meter.
One cubic meter can hold 6.3 barrels of oil. Thus, the costs are little less than 6 EUR / barrel to construct the storage in a perfect location.
The problem in this approach is that the tunnel construction in "good ground requiring minimal support" can't be done anywhere (you need to find the good ground requiring minimal support first) and there is a delay in the construction of such a tunnel. So, temporarily due to a panic oil prices can go more negative than 6 EUR / barrel. But it won't be permanent. Eventually new storage will be found in the form of newly constructed tunnels.