There's a club that sells paid memberships. In the application, it says upfront that the club can cancel memberships at any time, without reason and without giving a refund.
Is this really an enforceable contract? Is it possible it could be unconscionable that one side has essentially 100% power and the other side has no recourse?
Two essential elements of a contract are that each part gives something of value (such as money, or whatever the club has to offer), and each part has an obligation (the member must give money and do certain things, the club must allow the member to use the facility). Without these elements there is no contract, meaning that the "member" has no obligation to pay (and the courts will then order the return of the membership payment). If indeed the club can revoke membership at any time and for no cause, there is no contract and no legal justification for the club to keep your money.
The status of social clubs is not entirely clear, and it's not clear whether "membership" is a contractual matter, however an organization like Elks is a corporation, and state laws regulate expulsion of members of a corporation. They will typically require termination that is fair and reasonable and is carried out in good faith, as discussed in this article. In the cited cases where a person was expelled, it was for violation of the rules. It is unlikely that corporate laws of any state allow unilateral expulsion of a member for no cause.
If the OP is asking about a health club; it's not entirely clear what type of club is meant:
It is not "unconscionable that one side has essentially 100% power." The potential member of the club has 100% power to not sign the contract. The application contract will have stipulations about membership terms and refunds; it's the potential members' responsibility to read and agree before signing. The membership terms of the contract cannot violate state or federal law; if they do, the contract is void.
If the club member has discovered there are no membership refunds after signing, there's no right to cancel contracts in most states, unless in some cases it is for real estate or a mortgage, and then some states have a "cooling off" period where the contract can be cancelled. Where the purchase is $25 or more, federal law allows for the right to cancel for a full refund until midnight of the third business day after the sale, but that "rule applies to sales at the buyer's home or workplace, at facilities rented by the seller on a temporary basis, or at locations otherwise away from the seller's normal retail location." See https://www.findlaw.com/consumer/consumer-transactions/customer-returns-and-refund-laws-by-state.html
In Oregon, if the membership contract is with a gym, fitness facility, tanning salon or health spa, the member has the right to cancel within three days without penalty. Beyond that time frame, the contract is valid, and the business can cancel a membership and not give a refund if the contract terms allow it. See https://www.doj.state.or.us/consumer-protection/sales-scams-fraud/fitness-clubs-health-spas/
