How to save user inputs like in a database?

If you only need to save the element in a short term way you can use javascript and the HTML5 data element to save the email as an element of the current page. It is very temporary storage, but is the best you're going to get.

Edit: Here's how you can do this using jQuery based javascript.

HTML:

<input type="text" id="email">
<input type="button" id="emailButton">
<div id="data_div"></div>

jQuery Javascript:

function retrieveEmail() {
    var email = $('#data_div').data("email");
    // do something with the email variable as needed here
    // here's an example of retrieving it to send it to the server
    var paramStr = "email=" + email;

    $.ajax({
        url: './your_server_file_here',
        dataType: 'json',
        data: paramStr,
        success: function(data) {
           callBack(data);
    });
}

function callBack(data) {
    // do something with information passed back from the server after you sent the data to the server (you didn't say you needed to do this, but here's where it should be done)
}

function storeEmail() {
    var email = $('#email').val();
    $('#data_div').data("email", email);
}

$(document).ready( function() {
    $('#emailButton').click( function() {
        storeEmail();
    });
});

Edit: I know you already accepted this answer, but it struck me that HTML5 also includes another way to do this and it may provide the increase in power and flexibility that you're looking for. The HTML5 storage element can do pretty much the same thing as the data element except that it can persist and be accessed by other pages from the same domain either until the browser is closed (unlimited amount of data in the sessionStorage) or indefinitely (5mb of data in the localStorage). For implementation details and a greater understanding see here.

Be advised though, the HTML5 storage element is only available in HTML5 compatible browsers and should not be used if you fear your user base won't be using a modern browser. The HTML5 Data element will work even in older browsers, so it does have that advantage.