Is there demand for a service that sells database of contact information of varied sectors and geographies? If so, how do I market this product?

There are a lot of players in this market. Many that integrate with all the major CRMs and append and clean data.

I would say I get at least 1 email a day from a data vendor and I have been on countless demos and calls and because my client is so niche they never have more 5-10 records even viable for me.

I am sure for larger companies with broader markets this works well. They all tend to have very aggressive sales outreach. I have one company that has called me every 3 months for the past 2 years. At this point hes basically just asking about my health and the weather :)

Yes, there is demand. Depending on the quality and quantity of data as well as the price point, I and many of my colleagues would be potential customers.

Most of the people I see attempting to monetize a large database (whether of contact info, historical records, or dictionaries) start by putting up a website that offers a limited amount of queries for free.

Depending on the type of data, they'll either monetize the site with ads or else offer upsells for visitors who require more information. Some sites obscure part of the data. Some impose a query limit. They sell subscriptions rather than comprehensive downloads.

If I were you, I'd build a site and set up a tiered membership service rather than knocking on doors selling the database in its entirety. That way you maintain control, generate recurring revenue, and build a larger non-paying audience that will perform word-of-mouth marketing for free.

I understand that database may add value to good prospects. But the toughest challenge would be to find one. As pointed out by others, marketing approach should be bang on, apart from price point. But what would be the price point??
Do in depth research of competitors and overall market demand.

As far as I can understand from your business question, you are talking about database marketing. Database marketing is a form of direct marketing that uses databases of customers to generate targeted lists for direct marketing communications. The more useful data available, the more effective the campaign. Such data comes from several sources. Many businesses collect data as part of a typical business transaction. For example, since finance and insurance companies already must collect name, address, and other information for a sale, it takes little extra effort to retain this information in a database. Additional data comes in from customer service, which can keep a record of all their customer communications.
Besides if you do have any questions give me a call: https://clarity.fm/joy-brotonath