Is it possible to dynamically compile and execute C# code fragments?

The best solution in C#/all static .NET languages is to use the CodeDOM for such things. (As a note, its other main purpose is for dynamically constructing bits of code, or even whole classes.)

Here's a nice short example take from LukeH's blog, which uses some LINQ too just for fun.

using System;
using System.Collections.Generic;
using System.Linq;
using Microsoft.CSharp;
using System.CodeDom.Compiler;

class Program
{
    static void Main(string[] args)
    {
        var csc = new CSharpCodeProvider(new Dictionary<string, string>() { { "CompilerVersion", "v3.5" } });
        var parameters = new CompilerParameters(new[] { "mscorlib.dll", "System.Core.dll" }, "foo.exe", true);
        parameters.GenerateExecutable = true;
        CompilerResults results = csc.CompileAssemblyFromSource(parameters,
        @"using System.Linq;
            class Program {
              public static void Main(string[] args) {
                var q = from i in Enumerable.Range(1,100)
                          where i % 2 == 0
                          select i;
              }
            }");
        results.Errors.Cast<CompilerError>().ToList().ForEach(error => Console.WriteLine(error.ErrorText));
    }
}

The class of primary importance here is the CSharpCodeProvider which utilises the compiler to compile code on the fly. If you want to then run the code, you just need to use a bit of reflection to dynamically load the assembly and execute it.

Here is another example in C# that (although slightly less concise) additionally shows you precisely how to run the runtime-compiled code using the System.Reflection namespace.

You can compile a piece C# of code into memory and generate assembly bytes with Roslyn. It's already mentioned but would be worth adding some Roslyn example for this here. The following is the complete example:

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Reflection;
using Microsoft.CodeAnalysis;
using Microsoft.CodeAnalysis.CSharp;
using Microsoft.CodeAnalysis.Emit;

namespace RoslynCompileSample
{
    class Program
    {
        static void Main(string[] args)
        {
            // define source code, then parse it (to the type used for compilation)
            SyntaxTree syntaxTree = CSharpSyntaxTree.ParseText(@"
                using System;

                namespace RoslynCompileSample
                {
                    public class Writer
                    {
                        public void Write(string message)
                        {
                            Console.WriteLine(message);
                        }
                    }
                }");

            // define other necessary objects for compilation
            string assemblyName = Path.GetRandomFileName();
            MetadataReference[] references = new MetadataReference[]
            {
                MetadataReference.CreateFromFile(typeof(object).Assembly.Location),
                MetadataReference.CreateFromFile(typeof(Enumerable).Assembly.Location)
            };

            // analyse and generate IL code from syntax tree
            CSharpCompilation compilation = CSharpCompilation.Create(
                assemblyName,
                syntaxTrees: new[] { syntaxTree },
                references: references,
                options: new CSharpCompilationOptions(OutputKind.DynamicallyLinkedLibrary));

            using (var ms = new MemoryStream())
            {
                // write IL code into memory
                EmitResult result = compilation.Emit(ms);

                if (!result.Success)
                {
                    // handle exceptions
                    IEnumerable<Diagnostic> failures = result.Diagnostics.Where(diagnostic => 
                        diagnostic.IsWarningAsError || 
                        diagnostic.Severity == DiagnosticSeverity.Error);

                    foreach (Diagnostic diagnostic in failures)
                    {
                        Console.Error.WriteLine("{0}: {1}", diagnostic.Id, diagnostic.GetMessage());
                    }
                }
                else
                {
                    // load this 'virtual' DLL so that we can use
                    ms.Seek(0, SeekOrigin.Begin);
                    Assembly assembly = Assembly.Load(ms.ToArray());

                    // create instance of the desired class and call the desired function
                    Type type = assembly.GetType("RoslynCompileSample.Writer");
                    object obj = Activator.CreateInstance(type);
                    type.InvokeMember("Write",
                        BindingFlags.Default | BindingFlags.InvokeMethod,
                        null,
                        obj,
                        new object[] { "Hello World" });
                }
            }

            Console.ReadLine();
        }
    }
}

Others have already given good answers on how to generate code at runtime so I thought I would address your second paragraph. I have some experience with this and just want to share a lesson I learned from that experience.

At the very least, I could define an interface that they would be required to implement, then they would provide a code 'section' that implemented that interface.

You may have a problem if you use an interface as a base type. If you add a single new method to the interface in the future all existing client-supplied classes that implement the interface now become abstract, meaning you won't be able to compile or instantiate the client-supplied class at runtime.

I had this issue when it came time to add a new method after about 1 year of shipping the old interface and after distributing a large amount of "legacy" data that needed to be supported. I ended up making a new interface that inherited from the old one but this approach made it harder to load and instantiate the client-supplied classes because I had to check which interface was available.

One solution I thought of at the time was to instead use an actual class as a base type such as the one below. The class itself can be marked abstract but all methods should be empty virtual methods (not abstract methods). Clients can then override the methods they want and I can add new methods to the base class without invalidating existing client-supplied code.

public abstract class BaseClass
{
    public virtual void Foo1() { }
    public virtual bool Foo2() { return false; }
    ...
}

Regardless of whether this problem applies you should consider how to version the interface between your code base and the client-supplied code.